research article time consistent strategies for mean

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 709129 16 pageshttpdxdoiorg1011552013709129

Research ArticleTime Consistent Strategies for Mean-VarianceAsset-Liability Management Problems

Hui-qiang Ma12 Meng Wu3 and Nan-jing Huang2

1 School of Economics Southwest University for Nationalities Chengdu Sichuan 610041 China2Department of Mathematics Sichuan University Chengdu Sichuan 610064 China3 Business School Sichuan University Chengdu Sichuan 610064 China

Correspondence should be addressed to Meng Wu shancherishhotmailcom

Received 28 May 2013 Accepted 6 August 2013

Academic Editor Cheng Shao

Copyright copy 2013 Hui-qiang Ma et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper studies the optimal time consistent investment strategies in multiperiod asset-liability management problems undermean-variance criterion By applying time consistent model of Chen et al (2013) and employing dynamic programming techniquewe derive two-time consistent policies for asset-liability management problems in a market with and without a riskless assetrespectively We show that the presence of liability does affect the optimal strategy More specifically liability leads a parallel shiftof optimal time-consistent investment policy Moreover for an arbitrarily risk averse investor (under the variance criterion) withliability the time-diversification effects could be ignored in a market with a riskless asset however it should be considered in amarket without any riskless asset

1 Introduction

By using variance as a risk measure Markowitz [1] proposedthe classic mean-variance portfolio selection model whichhas become the theoretical foundation of modern financetheory and has been extended in several directions Oneof the main extensions for portfolio selection is to studythe optimal policy in a multiperiod setting For exampleLi and Ng [2] and Zhou and Li [3] employed an embed-ding technique to derive analytical solutions to multiperiodand continuous-time mean-variance models respectively Inmost of the studies in the multiperiod environment therehas been a common assumption that an investor has nolong-term liability However in reality many investmentinstitutions (eg pension funds insurance company andbanks) have paid great attention to their portfolios whiletaking into account their liabilities Further it has beenshown by Kell and Muller [4] and Sharpe and Tint [5] thatliability does affect the optimal policy More specifically in asingle-period setting liability leads a parallel shift of mean-variance optimal investment policy and affects the mean-variance efficient frontier Due to both theoretical interestand practical importance of asset-liability management the

research on mean-variance asset-liability management hasattracted recent attentions For example among othersLeippold et al [6] considered the multiperiod asset-liabilitymanagement problem where the liability is exogenous andfixed and derived an analytical optimal policy and an efficientfrontier Further they extended their research to the casewhere the liability is endogenous and controllable in [7] Chiuand Li [8] and Xie et al [9] studied continuous-time mean-variance asset-liability management problems respectivelyFurthermore Xie [10] studied mean-variance model withstochastic liability in a Markovian regime switching financialmarket and Zeng and Li [11] investigated asset-liabilitymanagement problem in a jump diffusion market

In most of the literature the popular approaches of deal-ing with dynamic mean-variance asset-liability managementproblem are embedding technique which was developed byLi and Ng [2] and dual method However since the iterated-expectation property does not hold for the variance operatorthe optimal asset-liability management policy (called pre-commitment strategy) derived by both the approaches doesnot satisfy Bellmanrsquos optimality principle and is time incon-sistentThemain reason is that the precommitment strategyfor time interval [119905 119879] computed at time 119905will not necessarily

2 Mathematical Problems in Engineering

coincide with the strategy which computed at time 119905 + Δ119905As a result at time 119905 + Δ119905 the strategy computed at time 119905

will not be implemented by the investor unless there existssome commitment mechanism Strotz [12] first formalizedtime inconsistency and pointed out that the conflict couldbe solved by a time-consistent strategy Very recently muchmore scholars have paid their attentions on constructinga time-consistent mean-variance portfolio choice Amongothers Basak and Chabakauri [13] provided a fully analyt-ical characterization of the optimal time-consistent mean-variance portfolio within a general incomplete market econ-omy Wang and Forsyth [14] developed a numerical schemefor determining the optimal asset allocation strategy for time-consistent continuous time mean-variance optimization Byallowing the trade-off between the mean and the variance ofthe terminal wealth varying over time Cui et al [15] proposedaweak time consistency to comparewith Bellmanrsquos optimalityprinciple and derived an optimal mean-variance portfoliostrategy In all the literature mentioned above the studiesmainly referred to Bellmanrsquos optimality principle Howeverthe requirement (for short REQ) that local optimum isalso globally optimum is not necessarily satisfied whichis an essential requirement in solving the relevant optimalportfolio problem by the dynamic programming techniqueIn order to make up this shortfall Chen et al [16] by using atime consistent dynamic risk measure proposed a separabledynamic mean-variance model and showed that the relevantoptimal investment policy satisfies not only the Bellmanrsquosoptimality principle but also the REQ

Although asset-liability management is an importantissue in modern finance theory the time-consistent asset-liability management problem has not attracted enoughattention Recently Li et al [17] reported the time-consistentasset-liability management problem in the continuous-timesetting They employed Basak and Chabakaurirsquos [13] modelto study the continuous-time asset-liability managementproblem They derived the time-consistent optimal strategyand showed that the time-consistent efficient frontier withliability is below that without liability However the derivedtime-consistent policy does not satisfy the REQ As the afore-mentioned importance of the REQ in this paper we employthemodel of Chen et al [16] to analysis themultiperiod asset-liability management problem We derive time-consistentoptimal investment policies in a market with and withouta riskless asset respectively After comparing the optimaltime-consistent policies withmyopic strategies we show thatfor an arbitrarily risk averse investor if there is a risklessasset in the market the time-diversification effects arisingfrommultiperiod optimization can be ignored otherwise theeffects should be considered

This paper proceeds as follows In the next sectionwe formulate a time-consistent asset-liability managementmodel In Section 3 we derive the time-consistent optimalpolicy for a market without riskless asset Section 4 derivesthe time-consistent optimal policy for a market with bothriskless and risky assets Section 5 performs numericalexamples to illustrate our results The paper is concluded inSection 6

2 Model Formulation

Throughout this paper we assume that 119879 isin N is a fixed andfinite time horizon and trading only takes place at time 119896 =

0 1 119879 Let (ΩF 119901) be a probability space and let a 120590-fieldF

119905be the available information at time 119905

Consider a securitymarket consisting of one riskless assetand 119899 risky assets The return of the riskless asset at the 119905thinvestment period is assumed to be 119904

119905 The return of the 119894th

risky asset at the 119905th investment period is denoted by 119877119894

119905

and the relevant random return vector is denoted by 119877119905=

(1198771

119905 119877

119899

119905)1015840 which isF

119905+1-measurable

Consider an investor with an initial endowment 1199090and a

liability 1198970 We assume that the liability cannot be controlled

and denote by 119897119905the accumulative liability at time 119905 Let119876

119905be

the return of the liability at the 119905th investment period whichis F119905+1

-measurable It is clear that 119897119905+1

= 119876119905119897119905 Assume that

(1198771015840

119905 119876119905)1015840 119905 = 0 119879minus1 are statistically independentDenote

the expected return vector by 120583119905= 119864(119877

119905) and the variance-

covariancematrix by Σ119905= (Cov(119877119894

119905 119877119895

119905))119899times119899

which is assumedto be positive definite throughout this paper It is clear that119864(1198771199051198771015840

119905) = Σ119905+ 1205831199051205831015840

119905is also positive definite Denote by Σ

0

119905=

(Cov(119876119905 1198771

119905) Cov(119876

119905 119877119899

119905))1015840 the covariance vector The

investor begins hisher investment at time 0 and invests thecash amount119906119894

119905in the 119894th risky asset at the beginning of the 119905th

investment period where 119905 = 0 119879 minus 1 119906119905= (1199061

119905 119906

119899

119905)1015840

is called an investment decision at the 119905th investment periodand 119906 = (119906

0 119906

119879minus1) is an investment policy during the

entire investment horizonWe denote the wealth and the sur-plus of the investor at time 119905 by 119909

119905and 119878119905= 119909119905minus119897119905 respectively

A single-period conditional risk mapping is defined as

120588119905(119878119905+1

| F119905) = Var

119905(119878119905+1

) minus 120582119905119864119905(119878119905+1

) (1)

where 119864119905(119878119905+1

) = 119864(119878119905+1

| F119905) and Var

119905(119878119905+1

) = Var(119878119905+1

|

F119905) are the single-period conditional expectation of 119878

119905+1and

its conditional variance respectively Tradeoff parameter 120582119905

as a constant which is defined on [0infin] is a weight whichpresents the relative importance of expected profit comparedto the risk Note that if 120582

119905= 0 then the investor is arbitrary

risk averse who only focus on the risk if 120582119905= infin then the

investor is risk neutral who only concerns maximizing theirexpected profit if 120582

119905isin (0infin) the investor is risk averse who

considers both expected profit and the risk in his decisionIn multiperiod portfolio selection two main optimal

investment policies are the myopic and time-consistentstrategiesMyopic Strategy is a strategy whereby at each time119905 the investor determines their optimal investment decisionassuming the instantaneous moments of assets returns willremain fixed at their current values for the remainder of theinvestment horizon More specifically for any 119905 isin 0 119879 minus

1 the myopic strategy 119906119898119910

119905is a solution to the following

problem

min119906119905isinΠ119905

Var119905(119878119905+1

) minus 120582119905119864119905(119878119905+1

) (2)

where Π119905is a set of all permit policies at time 119905 Intuitively

a myopic investor only cares about the mean and variance ofthe surplus at the current period

Mathematical Problems in Engineering 3

To measure the total risk of an investor amongmultiperi-ods (after time 119905) we employ a separable expected conditionalmapping which is defined as (see Chen et al [16])

119879

sum119895=119905+1

119864119905(120588119895minus1

(119878119895| F119895minus1

)) (3)

which reflects all the risk in the future Following thisassumption a separable dynamic mean-variance problem isdefined as

min119906isinΠ

119879

sum119895=1

119864 (120588119895minus1

(119878119895| F119895minus1

)) (4)

where Π is a set of all permit policies The optimal policyof problem (4) which satisfies both Bellmanrsquos optimalityprinciple and requirement REQ is called Time-ConsistentStrategy Note that both Bellmanrsquos optimality principle andrequirement REQ could be proved by following the method-ology of Chen et al [16]Thus problem (4) can be recursivelysolved by the dynamic programming technique Applying theiterated-expectation property of the expected operator thatis 119864(119864(sdot | F

119895) | F119896) = 119864(sdot | F

119896) for 119895 gt 119896 we have

119879

sum119895=1

119864 (120588119895minus1

(119878119895| F119895minus1

))

= 1205880(1198781) + 119864 (120588

1(1198782)

+ 1198641(1205882(1198783) + sdot sdot sdot

+ 119864119879minus3

(120588119879minus2

(119878119879minus1

)

+119864119879minus2

(120588119879minus1

(119878119879))) ))

(5)(for more details see Chen et al [16]) Then problem (4) isequivalent tomin119906isinΠ

1205880(1198781)

+ 119864 (1205881(1198782) + 1198641(1205882(1198783)

+ sdot sdot sdot + 119864119879minus3

(120588119879minus2

(119878119879minus1

)

+119864119879minus2

(120588119879minus1

(119878119879))) ))

(6)It follows from Bellmanrsquos optimality principle and (6) that(4) is equivalent to find an optimal strategy to satisfy thefollowing problem

min1199060

(1205880(1198781)

+ 119864min1199061

(1205881(1198782)

+ 1198641min1199062

(1205882(1198783) + sdot sdot sdot

+ 119864119879minus3

min119906119879minus2

(120588119879minus2

(119878119879minus1

)

+119864119879minus2

min119906119879minus1

(120588119879minus1

(119878119879))) )))

(7)

We solve this problem in the following sections In order todiscuss the impact of riskless asset the market is consideredin two cases with and without riskless asset We demonstratethese results in Sections 3 and 4 respectively

3 Time Consistent Optimal Strategy withoutRiskless Asset

Consider a market consisting of only 119899 risky assets andassume that the wealth process 119909

119905is in a self-financing

fashion We list the notations of this section in Table 1 Thewealth process 119909

119905could be described as follows

119909119905+1

= 1198771015840

119905119906119905 1198681015840119906119905= 119909119905 119905 = 0 1 119879 minus 1 (8)

where 119868 = (1 1)1015840isin R119899 In this setting problem (6) can

be written as follows

min 1205880(1198781) + 1198640(1205881(1198782)

+ 1198641(1205882(1198783) + sdot sdot sdot

+ 119864119879minus3

(120588119879minus2

(119878119879minus1

)

+119864119879minus2

(120588119879minus1

(119878119879))) ))

st 1198771015840

119905119906119905= 119909119905+1

1198681015840119906119905= 119909119905

119897119905+1

= 119876119905119897119905

119878119905+1

= 119909119905+1

minus 119897119905+1

119905 = 0 1 119879 minus 1

(9)

Since Σ1

119879minus1is positive definite we have 119886

119879minus1gt 0 Further

it follows from the nonnegative definiteness of 119864(119877119879minus1

1198771015840

119879minus1)

that Σ1

119879minus2is also positive definite and 119886

119879minus2gt 0 By using

mathematical induction we conclude that for any 119905 =

0 119879 minus 1 Σ1119905is also positive definite and 119886

119905gt 0

By applying Bellmanrsquos optimality principle the time-consistent optimal investment policy of problem (9) is givenin the following theorem

Theorem 1 The time-consistent optimal investment policy ofproblem (9) is given by

119906lowast

119905=

(Σ1

119905)minus1

119868

119886119905

119909119905+

1205821

119905

2(Σ1

119905)minus1

(120583119905minus 119902119905119868)

+ (Σ1

119905)minus1

(Σ0

119905minus 120574119905119868 + 120574119905+1

119864 (119876119905119877119905)) 119897119905

for 119905 = 0 119879 minus 1

(10)


Table 1 Model notations of Section 3

For 119905 = 119879 minus 1 For 119905 = 119879 minus 2 01205821

119879minus1= 120582119879minus1

1205821

119905= 120582119905+ 1205821

119905+1119902119905+1

Σ1

119879minus1= Σ119879minus1

Σ1

119905= Σ119905+ 119864 (119877

1199051198771015840

119905)119886119905+1

119886119879minus1

= 1198681015840(Σ1

119879minus1)minus1

119868 119886119905= 1198681015840(Σ1

119905)minus1

119868

119887119879minus1

= 1198681015840(Σ1

119879minus1)minus1

120583119879minus1

119887119905= 1198681015840(Σ1

119905)minus1

120583119905

119888119879minus1

= 1205831015840

119879minus1(Σ1

119879minus1)minus1

120583119879minus1

119888119905= 1205831015840

119905(Σ1

119905)minus1

120583119905

119889119879minus1

= 1198681015840(Σ1

119879minus1)minus1

Σ0

119879minus1119889119905= 1198681015840(Σ1

119905)minus1

Σ0

119905

119902119879minus1

= 119887119879minus1

119886119879minus1

119902119905= 119887119905119886119905

120572119879minus1

= 119888119879minus1

minus 119902119879minus1

119887119879minus1

120572119905= 119888119905minus 119902119905119887119905

120574119879minus1

= 119889119879minus1

119886119879minus1

120574119905= 119889119905119886119905

120574119879minus1

= 120574119879minus1

120574119905= 120574119905+ 120574119905+1

119864(1198761199051198771015840

119905)(Σ1

119905)minus1

119868119886119905

1205731119879minus1

= 120582119879minus1

119864(119876119879minus1

) minus 120582119879minus1

(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1

1205731119905

= minus1205821

1199051205831015840

119905(Σ1

119905)minus1

(Σ0

119905minus 120574119905119868) minus 120582

1

119905120574119905+1

1205831015840

119905(Σ1

119905)minus1

119864(119876119905119877119905)

+ 120582119879minus1

119887119879minus1

120574119879minus1

+ (120582119905+ 1205731119905+1

)119864(119876119905)

1205732119879minus1

= Var(119876119879minus1

) minus (Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0

119879minus1+ 120574119879minus1

119889119879minus1

1205732119905

= [Var(119876119905) + 1205732119905+1

119864(1198762

119905)] + 119886

1199051205742

119905minus [Σ0

119905+ 120574119905+1

119864(119876119905119877119905)]1015840

sdot (Σ1

119905)minus1[Σ0

119905+ 120574119905+1

119864(119876119905119877119905)]

Proof When 119905 = 119879minus1 for given wealth 119909119879minus1

and liability 119897119879minus1

at the beginning of the (119879 minus 1)th period problem (9) can beexpressed as follows

min Var119879minus1

(119878119879) minus 120582119879minus1

119864119879minus1

(119878119879)

st 1198771015840

119879minus1119906119879minus1

= 119909119879

1198681015840119906119879minus1

= 119909119879minus1

119897119879= 119876119879minus1

119897119879minus1

119878119879= 119909119879minus 119897119879

(11)

Substituting the binding constraints into the objective func-tion we have

min 119881119879minus1

(119909119879minus1

119897119879minus1

)

= 1199061015840

119879minus1Σ1

119879minus1119906119879minus1

minus 2 [(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21199061015840

119879minus1] 119906119879minus1

+ Var (119876119879minus1

) 1198972

119879minus1+ 120582119879minus1

119864 (119876119879minus1

) 119897119879minus1

st 1198681015840119906119879minus1

= 119909119879minus1

(12)

which is a linear-quadratic program By using the Lagrangemultiplier technique and letting 120596

119879be the Lagrange multi-

plier the Lagrange function is defined as

119871 (119906119879minus1

) = 1199061015840

119879minus1Σ1

119879minus1119906119879minus1

minus 2 [(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] 119906119879minus1

+ Var (119876119879minus1

) 1198972

119879minus1+ 120582119879minus1

119864 (119876119879minus1

) 119897119879minus1

minus 120596119879(119909119879minus1

minus 1198681015840119906119879minus1

)

(13)

By using the first-order necessary optimality condition wehave

2Σ1

119879minus1119906119879minus1

minus 2(Σ0

119879minus1119897119879minus1

+120582119879minus1

2120583119879minus1

) + 120596119879119868 = 0 (14)

119909119879minus1

minus 1198681015840119906119879minus1

= 0 (15)

From (14) we can easily have

119906lowast

119879minus1= (Σ1

119879minus1)minus1

(Σ0

119879minus1119897119879minus1

+120582119879minus1

2120583119879minus1

) minus1

2120596119879(Σ1

119879minus1)minus1

119868

(16)

1198681015840119906lowast

119879minus1= 1198681015840(Σ1

119879minus1)minus1

(Σ0

119879minus1119897119879minus1

+120582119879minus1

2120583119879minus1

)

minus1

21205961198791198681015840(Σ1

119879minus1)minus1

119868

= 119889119879minus1

119897119879minus1

+120582119879minus1

2119887119879minus1

minus1

2119886119879minus1

120596119879

= 119909119879minus1

(17)

which implies that the Lagrange multiplier 120596119879is

120596119879= 2

119889119879minus1

119886119879minus1

119897119879minus1

+ 120582119879minus1

119887119879minus1

119886119879minus1

minus 21

119886119879minus1

119909119879minus1

= 2120574119879minus1

119897119879minus1

+ 120582119879minus1

119902119879minus1

minus 21

119886119879minus1

119909119879minus1

(18)

Substituting 120596119879into (16) we have

119906lowast

119879minus1=

(Σ1

119879minus1)minus1

119868

119886119879minus1

119909119879minus1

+ (Σ1

119879minus1)minus1

[Σ0

119879minus1minus 120574119879minus1

119868] 119897119879minus1

+120582119879minus1

2(Σ1

119879minus1)minus1

[120583119879minus1


119868]

(19)


Further by substituting 119906lowast

119879minus1into the objective function of

problem (12) (see Appendix A for more details) we have

119881lowast

119879minus1(119909119879minus1

119897119879minus1

) =1

119886119879minus1

1199092


119902119879minus1

119909119879minus1


119909119879minus1

119897119879minus1

+ 1205731119879minus1

119897119879minus1

+ 1205732119879minus1

1198972

119879minus1minus

(120582119879minus1

)2

4120572119879minus1

(20)

When 119905 = 119879 minus 2 for given wealth 119909119879minus2


at the beginning of the (119879 minus 2)th period the correspondingoptimal investment problem is given as follows

min Var119879minus2

(119878119879minus1

) minus 120582119879minus2

119864119879minus2

(119878119879minus1

)

+ 119864119879minus2

(119881lowast

119879minus1(119909119879minus1

119897119879minus1

))

st 1198771015840

119879minus2119906119879minus2

= 119909119879minus1

1198681015840119906119879minus2

= 119909119879minus2

119897119879minus1

= 119876119879minus2

119897119879minus2

119878119879minus1

= 119909119879minus1


(21)

It follows from (20) that

119864119879minus2

(119881lowast

119879minus1(119909119879minus1

119897119879minus1

))

=1

119886119879minus1

119864119879minus2

(1199092

119879minus1) minus 120582119879minus1

119902119879minus1

119864119879minus2

(119909119879minus1

)


119864119879minus2

(119909119879minus1

119897119879minus1

) + 1205731119879minus1

119864119879minus2

(119897119879minus1

)

+ 1205732119879minus1

119864119879minus2

(1198972

119879minus1) minus

(120582119879minus1

)2

4120572119879minus1

(22)

Substituting 119864119879minus2

(119881lowast

119879minus1(119909119879minus1

119897119879minus1

)) and the binding con-straints into the objective function of problem (21) we have

min 119881119879minus2

(119909119879minus2

119897119879minus2

)

st 1198681015840119906119879minus2

= 119909119879minus2

(23)

where

119881119879minus2

(119909119879minus2

119897119879minus2

) = 1199061015840

119879minus2Σ1

119879minus2119906119879minus2

minus 2 [(Σ0

119879minus2)1015840

+ 120574119879minus1

119864 (119876119879minus2

1198771015840

119879minus2)] 119897119879minus2

+1205821

119879minus2

21205831015840

119879minus2119906119879minus2

+ [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1

minus(120582119879minus1

)2

4120572119879minus1

(24)

By setting 120596119879minus1

be the Lagrange multiplier the Lagrangefunction for problem (23) is

119871 (119906119879minus2

) = 1199061015840

119879minus2Σ1

119879minus2119906119879minus2

minus 2Φ1015840

119879minus2119906119879minus2

+ Ψ119879minus2


(119909119879minus2

minus 1198681015840119906119879minus2

)

(25)

where

Φ119879minus2

= [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)] 119897119879minus2

+1205821

119879minus2

2120583119879minus2

Ψ119879minus2

= [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1


)2

4120572119879minus1

(26)

From the first-order necessary optimality condition we have

2Σ1

119879minus2119906119879minus2

minus 2Φ119879minus2

+ 120596119879minus1

119868 = 0 (27)

119909119879minus2

minus 1198681015840119906119879minus2

= 0 (28)

From (27) we have

119906lowast

119879minus2= (Σ1

119879minus2)minus1

Φ119879minus2

minus1

2120596119879minus1

(Σ1

119879minus2)minus1

119868 (29)

1198681015840119906lowast

119879minus2= 1198681015840(Σ1

119879minus2)minus1

Φ119879minus2

minus1

2120596119879minus1

1198681015840(Σ1

119879minus2)minus1

119868

= 1198681015840(Σ1

119879minus2)minus1

Φ119879minus2

minus1

2120596119879minus1

119886119879minus2

= 119909119879minus2

(30)

which implies

120596119879minus1

= 21198681015840(Σ1

119879minus2)minus1

Φ119879minus2

119886119879minus2

minus 21

119886119879minus2

119909119879minus2

(31)


into (29) we get

119906lowast

119879minus2=

(Σ1

119879minus2)minus1

119868

119886119879minus2

119909119879minus2

+ (Σ1

119879minus2)minus1

Φ119879minus2

minus1198681015840(Σ1

119879minus2)minus1

Φ119879minus2

119886119879minus2

(Σ1

119879minus2)minus1

119868

(32)

Taking Φ119879minus2

into account we have

119906lowast

119879minus2=

(Σ1

119879minus2)minus1

119868

119886119879minus2

119909119879minus2

+1205821

119879minus2

2(Σ1

119879minus2)minus1

(120583119879minus2


119868)

+ (Σ1

119879minus2)minus1

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868] 119897119879minus2

(33)


Substituting 119906lowast

119879minus2into (23) gives (see Appendix B for more

details)

119881lowast

119879minus2(119909119879minus2

119897119879minus2

) =1

119886119879minus2

1199092

119879minus2minus 1205821

119879minus2119902119879minus2

119909119879minus2


119909119879minus2

119897119879minus2

+ 1205731119879minus2

119897119879minus2

+ 1205732119879minus2

1198972

119879minus2


)2

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(34)

Next by using mathematical induction we show thatboth (10) and

119881lowast

119905(119909119905 119897119905) =

1

119886119905

1199092

119905minus 1205821

119905119902119905119909119905minus 2120574119905119909119905119897119905

+ 1205731119905119897119905+ 12057321199051198972

119905minus

119879minus1

sum119894=119905

(1205821

119894)2

4120572119894

(35)

hold Suppose that (10) and (35) hold for time 119905 119905+1 119879minus1At the beginning of (119905minus1)th period for given wealth 119909

119905minus1and

liability 119897119905minus1

the corresponding optimal investment problemis

min Var119905minus1

(119878119905) minus 120582119905minus1

119864119905minus1

(119878119905)

+ 119864119905minus1

(119881lowast

119905(119909119905 119897119905))

st 1198771015840

119905minus1119906119905minus1

= 119909119905

1198681015840119906119905minus1

= 119909119905minus1

119897119905= 119876119905minus1

119897119905minus1

119878119905= 119909119905minus 119897119905

(36)


119864119905minus1

(119881lowast

119905(119909119905 119897119905)) =

1

119886119905

119864119905minus1

(1199092

119905) minus 1205821

119905119902119905119864119905minus1

(119909119905)

minus 2120574119905119864119905minus1

(119909119905119897119905) + 1205731119905119864119905minus1

(119897119905)

+ 1205732119905119864119905minus1

(1198972

119905) minus

119879minus1

sum119894=119905

(1205821

119894)2

4120572119894

(37)


(119881lowast

119905(119909119905 119897119905)) and the binding constraints into

the objective function of problem (36) we have

min 119881119905minus1

(119909119905minus1

119897119905minus1

) = 1199061015840

119879minus1Σ1

119905minus1119906119905minus1

minus 2Φ1015840

119905minus1119906119905minus1

+ Ψ119905minus1

st 1198681015840119906119905minus1

= 119909119905minus1

(38)

where

Φ119905minus1

= [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] 119897119905minus1

+1205821

119905minus1

2120583119905minus1

Ψ119905minus1

= [Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] 1198972

119905minus1

+ [120582119905minus1

+ 1205731119905] 119864 (119876

119905minus1) 119897119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4120572119894

(39)

Letting 120596119905be the Lagrange multiplier the Lagrange function

for problem (38) is given by

119871 (119906119905minus1

) = 1199061015840

119879minus1Σ1

119905minus1119906119905minus1

minus 2Φ1015840

119905minus1119906119905minus1

+ Ψ119905minus1

minus 120596119905(119909119905minus1

minus 1198681015840119906119905minus1

)

(40)

It follows from the first-order necessary optimality conditionthat

2Σ1

119905minus1119906119905minus1


+ 120596119905119868 = 0

119909119905minus1

minus 1198681015840119906119905minus1

= 0

(41)

Thus we have

119906lowast

119905minus1= (Σ1

119905minus1)minus1

Φ119905minus1

minus1

2120596119905(Σ1

119905minus1)minus1

119868 (42)

1198681015840119906lowast

119905minus1= 1198681015840(Σ1

119905minus1)minus1

Φ119905minus1

minus1

21205961199051198681015840(Σ1

119905minus1)minus1

119868

= 1198681015840(Σ1

119905minus1)minus1

Φ119905minus1

minus1

2120596119905119886119905minus1

= 119909119905minus1

(43)

which implies

120596119905= 2

1198681015840(Σ1

119905minus1)minus1

Φ119905minus1

119886119905minus1

minus 21

119886119905minus1

119909119905minus1

(44)


119906lowast

119905minus1=

(Σ1

119905minus1)minus1

119868

119886119905minus1

119909119905minus1

minus1198681015840(Σ1

119905minus1)minus1

Φ119905minus1

119886119905minus1

(Σ1

119905minus1)minus1

119868

+ (Σ1

119905minus1)minus1

Φ119905minus1

=(Σ1

119905minus1)minus1

119868

119886119905minus1

119909119905minus1

+1205821

119905minus1

2(Σ1

119905minus1)minus1

(120583119905minus1


119868)

+ (Σ1

119905minus1)minus1

[Σ0


119868 + 120574119905119864 (119876119905minus1

119877119905minus1

)] 119897119905minus1

(45)


119905minus1into the objective function of problem (38)

(see Appendix C for more details) we have

119881lowast

119905minus1(119909119905minus1

119897119905minus1

) =1

119886119905minus1

1199092


119905minus1119902119905minus1

119909119905minus1


119909119905minus1

119897119905minus1

+ 1205731119905minus1

119897119905minus1

+ 1205732119905minus1

1198972

119905minus1minus

119879minus1

sum119894=119905minus1

(1205821

119894)2

4120572119894

(46)This completes the proof


Remark 2 If an investor does not have any liability that is119897119905equiv 0 for any 119905 isin 0 1 119879 minus 1 then the optimal time-

consistent investment strategy can be simplified as follows

119906lowast

119905=

(Σ1

119905)minus1

119868

119886119905

119909119905+

1205821

119905

2(Σ1

119905)minus1

(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1

(47)

which is exactly the same as that in [16] This implies thatthe result of Chen et al [16] is a special case of Theorem 1Therefore Theorem 1 generalizes their result

Corollary 3 If the returns of liability and risky assets areuncorrelated that is Σ0

119905= 0 for any 119905 isin 0 1 119879 minus 1 then

the optimal investment policy for problem (9) is

119906lowast

119905=

(Σ1

119905)minus1

119868

119886119905

119909119905+

1205821

119905

2(Σ1

119905)minus1

(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1

(48)

Proof Since Σ0119905= 0 it is easy to verify that 119889

119905= 1198681015840(Σ1

119905)minus1Σ0

119905=

0 120574119905= 119889119905119886119905= 0 and 120574

119905= 0 Substituting them into (10) gives

119906lowast

119905=

(Σ1

119905)minus1

119868

119886119905

119909119905+

1205821

119905

2(Σ1

119905)minus1

(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1

(49)

This completes the proof

Remark 4 After comparing Corollary 3 and Remark 2 it isquite clear that if the return of liability is uncorrelated withthat of risky asset then the liability does not affect the time-consistent optimal policy in a market without riskless asset

Remark 5 If the return of liability is correlated to those ofrisky assets then the occurrence of liability leads to a parallelshift of the optimal investment policy and the shift is

(Σ1

119905)minus1

(Σ0

119905minus 120574119905119868 + 120574119905+1

119864 (119876119905119877119905)) 119897119905 (50)

which depends on the current value of liability 119897119905 and the

covariance between the returns of liability and risky assetsΣ0

119905

Next we compare the time-consistent strategy with themyopic strategy in a market without riskless asset In such amarket problem (2) can be expressed as follows

min119906119905

Var119905(119878119905+1

) minus 120582119905119864119905(119878119905+1

)

st 1198771015840

119905119906119905= 119909119905+1

1198681015840119906119905= 119909119905

119897119905+1

= 119876119905119897119905

119878119905+1

= 119909119905+1

minus 119897119905+1

(51)

By using the same method in the proof ofTheorem 1 for time119879 minus 1 the myopic strategy is given by

119906119898119910

119905=

(Σ119905)minus1

119868

119886119905

119909119905+

120582119905

2(Σ119905)minus1

(120583119905minus 119902119905119868)

+ (Σ119905)minus1

(Σ0

119905minus 120574119905119868) 119897119905 119905 = 0 119879 minus 1

(52)

where 119886119905= 1198681015840(Σ119905)minus1119868 119902119905= 1198681015840(Σ119905)minus1120583119905119886119905 120574119905= 1198681015840(Σ119905)minus1Σ0

119905119886119905

It is clear that the difference between two strategies enters intoall of the three parts More specifically the following featureholds if the investor is arbitrarily risk averse that is 120582

119905rarr 0

then both the time consistent optimal strategy and myopicstrategy reduce to

119906lowast

119905=

(Σ1

119905)minus1

119868

119886119905

119909119905+ (Σ1

119905)minus1

(Σ0

119905minus 120574119905119868 + 120574119905+1

119864 (119876119905119877119905)) 119897119905

119905 = 0 119879 minus 1

119906119898119910

119905=

(Σ119905)minus1

119868

119886119905

119909119905+ (Σ119905)minus1

(Σ0

119905minus 120574119905119868) 119897119905 119905 = 0 119879 minus 1

(53)

respectively After comparing these two strategies we findthat if an investor is arbitrarily risk averse then heshe shouldconcern about the time-diversification effects arising frommultiperiod optimization

4 Time Consistent Optimal Strategy withRiskless Asset

In this section we consider a market which is consistingof one riskless asset and 119899 risky assets and assume that thewealth process 119909

119905is also in a self-financing fashion We list

the notations of this section in Table 2 The wealth process 119909119905

can be described as follows

119909119905= 1198751015840

119905minus1119906119905minus1

+ 119904119905minus1

119909119905minus1

(54)

where119875119905= 119877119905minus119904119905119868 In this setting problem (6) can be written

as followsmin 120588

0(1198781) + 1198640(1205881(1198782) + 1198641(1205882(1198783) + sdot sdot sdot

+119864119879minus3

(120588119879minus2

(119878119879minus1

) + 119864119879minus2

(120588119879minus1

(119878119879))) ))

st 119909119905+1

= 1198751015840

119905119906119905+ 119904119905119909119905

119897119905+1

= 119876119905119897119905

119878119905+1

= 119909119905+1

minus 119897119905+1

119905 = 0 1 119879 minus 1

(55)


Theorem 6 The optimal investment strategy of problem (55)is given by

119906lowast

119905= Σminus1

119905120593119905119897119905+

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (56)

where 120593119905= 119864(119875

119905119876119905) minus 119864(119875

119905)119864(119876119905) and 120598

119905= 119864(119875

119905)




119879minus1= 120582119879minus1

1205821

119905= 120582119905+ 1205821

119905+1119904119905+1

1205731119879minus1

= 120582119879minus1

[119864 (119876119879minus1

) minus 1205981015840

119879minus1(Σ119879minus1

)minus1

120593119879minus1

] 1205731119905

= (120582119905+ 1205731119905+1

) 119864(119876119905) minus 1205821

1199051205981015840

119905(Σ119905)minus1120593119905

1205732119879minus1

= Var(119876119879minus1

) minus 1205931015840


)minus1

120593119879minus1

1205732119905

= Var(119876119905) + 1205732119905+1

119864(1198762

119905) minus 1205931015840

119905(Σ119905)minus1120593119905

Proof When 119905 = 119879minus1 for givenwealth 119909119879minus1


at the beginning of the (119879minus1)th period problem (55) reducesto

min119906119879minus1

Var119879minus1

(119878119879) minus 120582119879minus1

119864119879minus1

(119878119879)

st 119909119879

= 119904119879minus1

119909119879minus1

+ 1198751015840

119879minus1119906119879minus1

119897119879

= 119876119879minus1

119897119879minus1

119878119879= 119909119879minus 119897119879

(57)


min119906119879minus1

119881119879minus1

(119909119879minus1

119897119879minus1

) (58)

where

119881119879minus1

(119909119879minus1

119897119879minus1

) = 1199061015840

119879minus1Σ119879minus1

119906119879minus1

minus 2(1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840

119879minus1)119906119879minus1

+ Var (119876119879minus1

) 1198972


119904119879minus1

119909119879minus1

+ 120582119879minus1

119864 (119876119879minus1

) 119897119879minus1

(59)

It is clear that problem (58) is an unconstrained convex pro-gram problem By using the first-order necessary optimalitycondition we have

119889 (119881119879minus1

(119909119879minus1

119897119879minus1

))

119889 (119906119879minus1

)= 2Σ119879minus1

119906119879minus1

minus 2(120593119879minus1

119897119879minus1

+120582119879minus1

2120598119879minus1

)

= 0

(60)

which implies

119906lowast

119879minus1= (Σ119879minus1

)minus1

[120593119879minus1

119897119879minus1

+120582119879minus1

2120598119879minus1

] (61)



gives (see Appendix D for more details)

119881lowast

119879minus1(119909119879minus1

119897119879minus1

) = 1205732119879minus1

1198972

119879minus1+ 1205731119879minus1

119897119879minus1


119904119879minus1

119909119879minus1


)2

4119872119879minus1

(62)

where119872119879minus1

= 1205981015840

119879minus1Σminus1

119879minus1120598119879minus1



at the beginning of the (119879 minus 2)th period the correspondingoptimal investment problem can be expressed as follows

min Var119879minus2

(119878119879minus1

) minus 120582119879minus2

119864119879minus2

(119878119879minus1

)

+ 119864119879minus2

(119881lowast

119879minus1(119909119879minus1

119897119879minus1

))

st 119909119879minus1

= 119904119879minus2

119909119879minus2

+ 1198751015840

119879minus2119906119879minus2

119897119879minus1

= 119876119879minus2

119897119879minus2

119878119879minus1

= 119909119879minus1


(63)


119864119879minus2

(119881lowast

119879minus1(119909119879minus1

119897119879minus1

)) = 1205732119879minus1

119864 (1198762

119879minus2) 1198972

119879minus2

+ 1205731119879minus1

119864 (119876119879minus2

) 119897119879minus2


119904119879minus1

119904119879minus2

119909119879minus2


119904119879minus1

1205981015840

119879minus2119906119879minus2


)2

4119872119879minus1

(64)


(119881lowast

119879minus1(119909119879minus1

119897119879minus1


min119906119879minus2

119881119879minus2

(119909119879minus2

119897119879minus2

) (65)


(119909119879minus2

119897119879minus2

) = 1199061015840


119906119879minus2

minus 2(1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840

119879minus2)119906119879minus2

+ [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

(66)

The first-order necessary optimality condition implies

119889 (119881119879minus2

(119909119879minus2

119897119879minus2

))

119889 (119906119879minus2

)= 2Σ119879minus2

119906119879minus2

minus 2(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= 0

(67)


Thus

119906lowast

119879minus2= (Σ119879minus2

)minus1

(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= (Σ119879minus2

)minus1

120593119879minus2

119897119879minus2

+1205821

119879minus2

2(Σ119879minus2

)minus1

120598119879minus2

(68)



(see Appendix E for more details) we have

119881lowast

119879minus2(119909119879minus2

119897119879minus2

) = 1205732119879minus2

1198972

119879minus2+ 1205731119879minus2

119897119879minus2

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

minus(1205821

119879minus2)2

4119872119879minus2

(69)


= 1205981015840


119879minus2120598119879minus2

Next by using mathematical induction we show that

both (56) and

119881lowast

119905(119909119905 119897119905) = 12057321199051198972

119905+ 1205731119905119897119905minus 1205821

119905119904119905119909119905minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

(70)

hold where 119872119894= 1205981015840

119894Σminus1

119894120598119894with 119894 = 0 1 119879 minus 1 Suppose

that (56) and (70) are true for time 119905 119905 + 1 119879 minus 1 At thebeginning of the (119905 minus 1)th period for given wealth 119909

119905minus1and



min Var119905minus1

(119878119905) minus 120582119905minus1

119864119905minus1

(119878119905) + 119864119905minus1

(119881lowast

119905(119909119905 119897119905))

st 119909119905= 119904119905minus1

119909119905minus1

+ 1198751015840

119905minus1119906119905minus1

119897119905= 119876119905minus1

119897119905minus1

119878119905= 119909119905minus 119897119905

(71)


119864119905minus1

(119881lowast

119905(119909119905 119897119905)) = 120573

2119905119864 (1198762

119905minus1) 1198972

119905minus1+ 1205731119905119864 (119876119905minus1

) 119897119905

minus 1205821

119905119904119905119904119905minus1

119909119905minus1

minus 1205821

1199051199041199051205981015840

119905minus1119906119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

(72)


(119881lowast


the objective function of problem (71) we havemin119906119905minus1

119881119905minus1

(119909119905minus1

119897119905minus1

) (73)

where

119881119905minus1

(119909119905minus1

119897119905minus1

) = 1199061015840


119906119905minus1

minus 2(1205931015840

119905minus1119897119905minus1

+1205821

119905minus1

21205981015840

119905minus1)119906119905minus1

+ [Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] 1198972

119905minus1

+ [120582119905minus1

119864 (119876119905minus1

) + 1205731119905119864 (119876119905minus1

)] 119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

(74)

The first-order necessary optimality condition gives

119889 (119881119905minus1

(119909119905minus1

119897119905minus1

))

119889 (119906119905minus1

)= 2Σ119905minus1

119906119905minus1

minus 2(120593119905minus1

119897119905minus1

+1205821

119905minus1

2120598119905minus1

)

= 0

(75)

which implies

119906lowast

119905minus1= Σminus1

119905minus1120593119905minus1

119897119905minus1

+1205821

119905minus1

2Σminus1

119905minus1120598119905minus1

(76)



(see Appendix F for more details) we have

119881lowast

119905minus1(119909119905minus1

119897119905minus1

) = 1205732119905minus1

1198972

119905minus1+ 1205731119905minus1

119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905minus1

(1205821

119894)2

4119872119894

(77)


Remark 7 From Theorem 6 it is clear that if there is ariskless asset in the market then the time-consistent optimalinvestment policy is wholly independent of the currentwealth 119909

119905 However Theorem 1 gives an opposite conclusion

This implies that the riskless asset does affect the optimalstrategy Therefore an investor should carefully select themarket they invested

Remark 8 If there is no liability that is 119897119905equiv 0 for any 119905 isin

0 1 119879 minus 1 then the time-consistent optimal investmentpolicy reduces to

119906lowast

119905=

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (78)

which is the same as that in [16] This implies that the resultof Chen et al [16] is a special case of Theorem 6 ThereforeTheorem 6 generalizes their result

Corollary 9 If the return of liability 119876119905 is uncorrelated with

those of risky assets 119877119905 that is Σ0

119905= 0 for any 119905 isin 0 1 119879minus

1 then the optimal policy for problem (55) is

119906lowast

119905=

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (79)

Proof Since Σ0119905= 0 it is easy to have

120593119905= 119864 (119875

119905119876119905) minus 119864 (119875

119905) 119864 (119876

119905)

= 119864 ((119877119905minus 119904119905119868)119876119905) minus 119864 (119877

119905minus 119904119905119868) 119864 (119876

119905)

= 119864 (119877119905119876119905) minus 119864 (119877

119905) 119864 (119876

119905)

= Σ0

119905

= 0

(80)


Substituting 120593119905= 0 into (56) gives

119906lowast

119905=

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (81)


Remark 10 After comparing Corollary 9 and Remark 8 it isquite clear that if the return of liability is uncorrelated withthose of risky assets then the occurrence of liability doesnot affect the time-consistent optimal investment policy in amarket with riskless asset

Remark 11 If the return of liability is correlated with those ofrisky assets then the occurrence of liability leads to a parallelshift of the optimal investment policy and the shift is

Σminus1

119905120593119905119897119905 (82)

which depends on the current value of the liability

Now we compare the time-consistent policy with themyopic strategy in a market with a riskless asset In such amarket problem (2) can be further expressed as follows

min119906119905

Var119905(119878119905+1

) minus 120582119905119864119905(119878119905+1

)

st 119909119905+1

= 119904119905119909119905+ 1198751015840

119905119906119905

119897119905+1

= 119876119905119897119905

119878119905+1

= 119909119905+1

minus 119897119905+1

(83)

By using the samemethod in the proof ofTheorem 6 for time119879 minus 1 the myopic strategy is given by

119906119898119910

119905= Σminus1

119905120593119905119897119905+

120582119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (84)

Differentwith themarketwithout riskless asset the differencebetween the time-consistent optimal strategy and the myopicstrategy only enters in the part depending on the risk aversion120582119905 Further the following feature holds if an investor is

arbitrarily risk averse that is 120582119905

rarr 0 then both the time-consistent optimal investment policy and themyopic strategyreduce to

119906lowast

119905= Σminus1

119905120593119905119897119905 119905 = 0 119879 minus 1 (85)

This implies that if an investor is arbitrarily risk averse thenheshe could ignore the time-diversification effects arisingfrom multiperiod optimization Further if the investor doesnot have any liability then both two strategies suggest thatheshe should leave the market

Remark 12 After comparing the results of these two differentmarkets we find that for an arbitrarily risk averse investor ifthere is a riskless asset in the market the time-diversificationeffects could be ignored otherwise the effects should beconsidered

Table 3 Time-consistent strategies with and without liability for1205820= 1205821= 05

119905Time-consistent strategy with

liability 119906lowast

119905

Time-consistent strategywithout liability

lowast

119905

0 (

11941198970+ 0903119909

0+ 3736

01071198970minus 0066119909

0+ 0203

minus05771198970+ 0163119909

0minus 3939

) (

09031199090+ 3736

minus00661199090+ 0203

01631199090minus 3939

)

1 (

minus03721198971+ 1018119909

1+ 1743

01981198971minus 0060119909

1+ 0095

minus06331198971+ 0042119909

1minus 1838

) (

10181199091+ 1743

minus00601199091+ 0095

00421199091minus 1838

)

Table 4 Investment strategies in a market without riskless asset for1205820= 1205821= 05

119905 Time-consistent strategy 119906lowast

119905Myopic strategy 119906

my119905

0 (

09031199090+ 1194119897

0+ 3736

minus00661199090+ 0107119897

0+ 0203

01631199090minus 0577119897

0minus 3939

) (

10181199090+ 0449119897

0+ 1743

minus00601199090+ 0150119897

0+ 0095

00421199090minus 0599119897

0minus 1838

)

1 (

10181199091minus 0372119897

1+ 1743

minus00601199091+ 0198119897

1+ 0095

00421199091minus 0633119897

1minus 1838

) (

10181199091+ 0449119897

1+ 1743

minus00601199091+ 0150119897

1+ 0095

00421199091minus 0599119897

1minus 1838

)

5 Numerical Illustration

In this section we present numerical examples to gaininsights regarding the impact of time diversification and ofliability on the optimal time-consistent strategies To makeit easy to analysis we assume 119879 = 2 and all parametersat different periods are the same Considering a marketwith three risky assets whose corresponding expected returnvector and the variance-covariance matrices are given as 120583

119905=

(1162 1246 1228) and

Σ119905= (

00146 00187 00145

00187 00854 00104

00145 00104 00289

) (86)

respectively The expected return of the liability 119864(119876119905) is

1136 the corresponding variance Var(119876119905) is 001 and the

covariance vector Σ0119905is (00006 00149 00050)1015840

Table 3 illustrates how the time-consistent strategydepends on the liability From Table 3 if an investor has aliability then heshe could adjust their investment strategywhich results in a parallel shift of the optimal time-consistentstrategyThus the investor should take account for the impactof liability

Tables 4 and 5 show the time-consistent strategy and themyopic strategy in a market without riskless asset for 120582

119905=

05 and 120582119905

= 0 respectively In Table 4 we find that thetwo strategies are different and the difference between thementers into all of the three parts Table 5 figures out that thetwo strategies are still very different even if the investor isarbitrarily risk averse Further Tables 4 and 5 imply that theinvestor can not ignore the time diversification effects in amarket without riskless asset

Next we consider a market consisting of both riskyassets and a riskless asset Suppose that the return of the





my119905

0 (

09031199090+ 1194119897

0

minus00661199090+ 0107119897

0

01631199090minus 0577119897

0

) (

10181199090+ 0449119897

0

minus00601199090+ 0150119897

0

00421199090minus 0599119897

0

)

1 (

10181199091minus 0372119897

1

minus00601199091+ 0198119897

1

00421199091minus 0633119897

1

) (

10181199091+ 0449119897

1

minus00601199091+ 0150119897

1

00421199091minus 0599119897

1

)

Table 6 Investment strategies in a market with a riskless asset for1205820= 1205821= 05



my119905

0 (

04441198970+ 7889

01501198970minus 0062

minus05991198970minus 3572

) (

04441198970+ 3867

01501198970minus 0031

minus05991198970minus 1751

)

1 (

04441198971+ 3867

01501198971minus 0031

minus05991198971minus 1751

) (

04441198971+ 3867

01501198971minus 0031

minus05991198971minus 1751

)




my119905

0 (

04441198970

01501198970

minus05991198970

) (

04441198970

01501198970

minus05991198970

)

1 (

04441198971

01501198971

minus05991198971

) (

04441198971

01501198971

minus05991198971

)

riskless asset 119904119905is 104 Then we have 120598

119905= 120583119905minus 119904119905119868 =

(0122 0206 0188)1015840 and120593

119905= Σ0

119905= (00006 00149 00050)

1015840Tables 6 and 7 show the time-consistent strategy and the

myopic strategy in a market with a riskless asset for 120582119905= 05

and 120582119905= 0 respectively From Table 6 it is clear that the two

strategies are different which is consistent with the results ina market without riskless asset However they may be exactlythe same if the investor is arbitrarily risk averse This impliesthat the investor who is arbitrarily risk averse can ignore thetime diversification effects

6 Conclusion

In this paper we consider the time-consistency of theoptimal asset-liability management policies in a market withand without a riskless asset respectively By employing thedynamic programming technique we give the optimal time-consistent investment policies After comparing the optimaltime-consistent policies withmyopic strategies we show thata risk averse investor should concern the time diversificationeffects Further an arbitrarily risk averse investor couldignore these effects in a market with a riskless asset

Appendices

A Proof of (20)

Firstly substituting 119906lowast

119879minus1into 119906

lowast

119879minus1

1015840Σ1

119879minus1119906lowast

119879minus1gives

119906lowast

119879minus1

1015840

Σ1


119879minus1=

119868

119886119879minus1

119909119879minus1

+ [Σ0


119868] 119897119879minus1

+120582119879minus1

2[120583119879minus1


119868]

1015840

(Σ1

119879minus1)minus1

times 119868

119886119879minus1

119909119879minus1

+ [Σ0


119868]

times119897119879minus1

+120582119879minus1

2[120583119879minus1


119868]

=1

119886119879minus1

1199092

119879minus1+ [Σ0


119868]1015840

times (Σ1

119879minus1)minus1

[Σ0


119868] 1198972

119879minus1

+ 120582119879minus1

[Σ0


119868]1015840

times (Σ1

119879minus1)minus1

[120583119879minus1


119868] 119897119879minus1

+(120582119879minus1

)2

4[120583119879minus1


119868]1015840

times (Σ1

119879minus1)minus1

[120583119879minus1


119868]

=1

119886119879minus1

1199092

119879minus1+ [(Σ

0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0

119879minus1

minus119886119879minus1

1205742

119879minus1] 1198972

119879minus1

+ [(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1


120574119879minus1

]

times 120582119879minus1

119897119879minus1

+(120582119879minus1

)2

4120572119879minus1

(A1)

Secondly by substituting 119906lowast

119879minus1into 2[(Σ

0

119879minus1)1015840

119897119879minus1

+

(120582119879minus1

2)1205831015840

119879minus1]119906lowast

119879minus1 we have

2 [(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] 119906lowast

119879minus1

=2

119886119879minus1

[(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1

119868119909119879minus1

+ 2 [(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1


times [Σ0


119868] 119897119879minus1

+ 120582119879minus1

[(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1

times [120583119879minus1


119868]

= 2120574119879minus1

119897119879minus1

119909119879minus1

+ 119902119879minus1

120582119879minus1

119909119879minus1

+ 2 [(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0


119889119879minus1

] 1198972

119879minus1

+ 2120582119879minus1

[(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1


120574119879minus1

] 119897119879minus1

+(120582119879minus1

)2

2120572119879minus1

(A2)

Thus119881lowast

119879minus1(119909119879minus1

119897119879minus1

)

=1

119886119879minus1

1199092


119902119879minus1

119909119879minus1


119909119879minus1

119897119879minus1

+ 120582119879minus1

[119864 (119876119879minus1

) minus (Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1

+119887119879minus1

120574119879minus1

] 119897119879minus1

+ [Var (119876119879minus1

) minus (Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0

119879minus1

+120574119879minus1

119889119879minus1

] 1198972

119879minus1minus

(120582119879minus1

)2

4120572119879minus1

=1

119886119879minus1

1199092


119902119879minus1

119909119879minus1


119909119879minus1

119897119879minus1

+ 1205731119879minus1

119897119879minus1

+ 1205732119879minus1

1198972

119879minus1minus

(120582119879minus1

)2

4120572119879minus1

(A3)

B Proof of (34)It is easy to verify that

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

=119889119879minus2

119886119879minus2

+ 120574119879minus1

1198681015840(Σ1

119879minus2)minus1

119864 (119876119879minus2

119877119879minus2

)

119886119879minus2


= 120574119879minus2

+ 120574119879minus1

1198681015840(Σ1

119879minus2)minus1

119864 (119876119879minus2

119877119879minus2

)

119886119879minus2


= 120574119879minus2


= 0

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1

[120583119879minus2


119868] =119887119879minus2

119886119879minus2


= 0

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

119868

= 119886119879minus2

120574119879minus2

+ 120574119879minus1

119864 (119876119879minus2

1198771015840

119879minus2) (Σ1

119879minus2)minus1

119868

119886119879minus2

= 119886119879minus2

120574119879minus2

(B1)

Firstly we compute 119906lowast

119879minus2

1015840Σ1


119879minus2 It follows from the

previous equalities that both coefficients of 119909119879minus2

and 119897119879minus2

119909119879minus2

in 119906lowast

119879minus2

1015840Σ1


119879minus2are 0 the coefficient of 1199092

119879minus2is

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1 119868

119886119879minus2

=1

119886119879minus2

(B2)

the coefficient of 119897119879minus2

is

1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

= 1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

minus 1205821

119879minus2120574119879minus2

1198681015840(Σ1

119879minus2)minus1

[120583119879minus2


119868]

= 1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

(B3)


is

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]1015840

sdot (Σ1

119879minus2)minus1

sdot [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

= [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]


1205742

119879minus2

(B4)

and the constant term is

(1205821

119879minus2)2

4[120583119879minus2


119868]1015840

(Σ1

119879minus2)minus1

[120583119879minus2


119868]

=(1205821

119879minus2)2

4120572119879minus2

(B5)

Secondly we compute Φ1015840


119879minus2 The coefficient of 1199092

119879minus2

is 0 the coefficient of term 119909119879minus2

is

1205821

119879minus2

2

1205831015840

119879minus2(Σ1

119879minus2)minus1

119868

119886119879minus2

=1205821

119879minus2119902119879minus2

2 (B6)



119909119879minus2

is 120574119879minus2


is

1205821

119879minus2

2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

(120583119879minus2


119868)

+1205821

119879minus2

21205831015840

119879minus2(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

(B7)


is

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

= [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)] minus 119886119879minus2

1205742

119879minus2

(B8)


(1205821

119879minus2

2)

2

1205831015840

119879minus2(Σ1

119879minus2)minus1

(120583119879minus2


119868) = (1205821

119879minus2

2)

2

120572119879minus2

(B9)

Then we can compute 119881lowast

119879minus2(119909119879minus2

119897119879minus2

) easily The coeffi-cient of 1199092

119879minus2is

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1 119868

119886119879minus2

=1

119886119879minus2

(B10)


isminus1205821119879minus2

119902119879minus2


119909119879minus2

is minus2120574119879minus2


is

[Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] + 119886

119879minus21205742

119879minus2

minus [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]

= 1205732119879minus2

(B11)


is

minus 1205821

119879minus21205831015840

119879minus2(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)]

= 1205731119879minus2

(B12)


minus1205822

119879minus1

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(B13)

Thus

119881lowast

119879minus2(119909119879minus2

119897119879minus2

) =1

119886119879minus2

1199092


119879minus2119902119879minus2

119909119879minus2


119909119879minus2

119897119879minus2

+ 1205731119879minus2

119897119879minus2

+ 1205732119879minus2

1198972

119879minus2minus

1205822

119879minus1

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(B14)

C Proof of (46)It is easy to verify that

1198681015840

119886119905minus1

(Σ1

119905minus1)minus1

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

=119889119905minus1

119886119905minus1

+ 120574119905

1198681015840(Σ1

119905minus1)minus1

119864 (119876119905minus1

119877119905minus1

)

119886119905minus1


= 120574119905minus1

+ 120574119905

1198681015840(Σ1

119905minus1)minus1

119864 (119876119905minus1

119877119905minus1

)

119886119905minus1


= 120574119905minus1


= 0

1198681015840

119886119905minus1

(Σ1

119905minus1)minus1

[120583119905minus1


119868] =119887119905minus1

119886119905minus1


= 0

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

119868

= 119886119905minus1

120574119905minus1

+ 120574119905

119864 (119876119905minus1

1198771015840

119905minus1) (Σ1

119905minus1)minus1

119868

119886119905minus1

= 119886119905minus1

120574119905minus1

(C1)


119905minus1

1015840Σ1




and 119897119905minus1

119909119905minus1

are 0 the coefficient of 1199092119905minus1

is

1198681015840

119886119905minus1

(Σ1

119905minus1)minus1 119868

119886119905minus1

=1

119886119905minus1

(C2)


is

1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

= 1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

minus 1205821

119905minus1120574119905minus1

1198681015840(Σ1

119905minus1)minus1

[120583119905minus1


119868]


= 1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

(C3)


is

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]1015840

sdot (Σ1

119905minus1)minus1

sdot [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

= [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] minus 119886119905minus1

1205742

119905minus1

(C4)


(1205821

119905minus1)2

4[120583119905minus1


119868]1015840

(Σ1

119905minus1)minus1

[120583119905minus1


119868] =(1205821

119905minus1)2

4120572119905minus1

(C5)




119905minus1is

0 the coefficient of 119909119905minus1

is

1205821

119905minus1

2

1205831015840

119905minus1(Σ1

119905minus1)minus1

119868

119886119905minus1

=1205821

119905minus1119902119905minus1

2 (C6)


119909119905minus1

is 120574119905minus1


is

1205821

119905minus1

2[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times (120583119905minus1


119868) +1205821

119905minus1

21205831015840

119905minus1(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

(C7)


is

[Σ0

119905minus1+ 120574119905119864(119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

= [Σ0

119905minus1+ 120574119905119864(119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] minus 119886119905minus1

1205742

119905minus1

(C8)


(1205821

119905minus1

2)

2

120583119905minus1

(Σ1

119905minus1)minus1

(120583119905minus1


119868) = (1205821

119905minus1

2)

2

120572119905minus1

(C9)

Then it is easy to compute 119881lowast

119905minus1(119909119905minus1

119897119905minus1

) The coefficientof 1199092

119905minus1is 1119886

119905minus1 the coefficient of 119909

119905minus1is minus120582

1

119905minus1119902119905minus1

thecoefficient of 119897

119905minus1119909119905minus1



is

[Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] + 119886

119905minus11205742

119905minus1

minus [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]

= 1205732119905minus1

(C10)


is

minus 1205821

119905minus11205831015840

119905minus1(Σ1

119905minus1)minus1

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

+ (120582119905minus1

+ 1205731119905) 119864 (119876

119905minus1) = 1205731119905minus1

(C11)

and the constant term is minus(14)sum119879minus1119894=119905minus1

(1205821

119894)2120572119894

Thus

119881lowast

119905minus1(119909119905minus1

119897119905minus1

) =1

119886119905minus1

1199092


119905minus1119902119905minus1

119909119905minus1


119909119905minus1

119897119905minus1

+ 1205731119905minus1

119897119905minus1

+ 1205732119905minus1

1198972

119905minus1minus

1

4

119879minus1

sum119894=119905minus1

(1205821

119894)2

120572119894

(C12)

D Proof of (62)


119879minus1into 119906

lowast

119879minus1

1015840Σ119879minus1

119906lowast


119906lowast

119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1= (1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840

119879minus1)

times (Σ119879minus1

)minus1

(120593119879minus1

119897119879minus1

+120582119879minus1

2120598119879minus1

)

= 1205931015840


)minus1

120593119879minus1

1198972

119879minus1+ 120582119879minus1

1205981015840

119879minus1


)minus1

120593119879minus1

119897119879minus1

+(120582119879minus1

)2

4119872119879minus1

2 (1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840

119879minus1)119906lowast

119879minus1= 2119906lowast

119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1

(D1)

Thus

119881lowast

119879minus1(119909119879minus1

119897119879minus1

) = minus119906lowast

119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1+ Var (119876

119879minus1) 1198972

119879minus1


119904119879minus1

119909119879minus1

+ 120582119879minus1

119864 (119876119879minus1

) 119897119879minus1

= [Var (119876119879minus1

) minus 1205931015840


)minus1

120593119879minus1

] 1198972

119879minus1

+ 120582119879minus1

[119864 (119876119879minus1

) minus 1205981015840


)minus1

120593119879minus1

]



119904119879minus1

119909119879minus1


)2

4119872119879minus1

= 1205732119879minus1

1198972

119879minus1+ 1205731119879minus1

119897119879minus1


119904119879minus1

119909119879minus1


)2

4119872119879minus1

(D2)


E Proof of (69)Taking 119906

lowast

119879minus2into account we have

119906lowast

119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2= (1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840

119879minus2)


)minus1

(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= 1205931015840


)minus1

120593119879minus2

1198972

119879minus2+ 1205821

119879minus21205981015840

119879minus2


)minus1

120593119879minus2

119897119879minus2

+(1205821

119879minus2)2

4119872119879minus2

2 1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840



119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2

(E1)Thus

119881lowast

119879minus2(119909119879minus2

119897119879minus2


119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2

+ [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

= 1205732119879minus2

1198972

119879minus2+ 1205731119879minus2

119897119879minus2

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

minus(1205821

119879minus2)2

4119872119879minus2

(E2)where1205731119879minus2

= (120582119879minus2

+ 1205731119879minus1

) 119864 (119876119879minus2

) minus 1205821

119879minus21205981015840


)minus1

120593119879minus2

1205732119879minus2

= Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2) minus 1205931015840


)minus1

120593119879minus2

(E3)

F Proof of (77)It follows from 119906

lowast

119905minus1that

119906lowast

119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1= (1205931015840

119879minus1119897119905minus1

+1205821

119905minus1

21205981015840

119879minus1)


)minus1

(120593119905minus1

119897119905minus1

+1205821

119905minus1

2120598119905minus1

)

= 1205931015840


)minus1

120593119905minus1

1198972

119905minus1+ 1205821

119905minus11205981015840

119879minus1


)minus1

120593119905minus1

119897119905minus1

+(1205821

119905minus1)2

4119872119905minus1

2 (1205931015840

119879minus1119897119905minus1

+1205821

119905minus1

21205981015840



119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1

(F1)

Thus

119881lowast

119905minus1(119909119905minus1

119897119905minus1


119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1

+ [Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] 1198972

119905minus1

+ [120582119905minus1

119864 (119876119905minus1

) + 1205731119905119864 (119876119905minus1

)] 119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

= 1205732119905minus1

1198972

119905minus1+ 1205731119905minus1

119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905minus1

(1205821

119894)2

4119872119894

(F2)

where

1205731119905minus1

= (120582119905minus1

+ 1205731119905) 119864 (119876

119905minus1) minus 1205821

119905minus11205981015840


)minus1

120593119905minus1

1205732119905minus1

= Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1) minus 1205931015840


)minus1

120593119905minus1

(F3)

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (71101099 11171237 11101069) andby the Construction Foundation of Southwest Universityfor Nationalities for the subject of Applied Economics(2011XWD-S0202)

References

[1] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952

[2] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000

[3] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000

[4] A Kell and H Muller ldquoEfficient portfolios in the asset liabilitycontextrdquo Astin Bulletin vol 25 pp 33ndash48 1995

[5] W F Sharpe andLG Tint ldquoLiabilities-a new approachrdquo Journalof Portfolio Management vol 16 pp 5ndash10 1990

[6] M Leippold F Trojani and P Vanini ldquoA geometric approachto multiperiod mean variance optimization of assets and liabil-itiesrdquo Journal of Economic Dynamics amp Control vol 28 no 6pp 1079ndash1113 2004

[7] M Leippold F Trojani and P Vanini ldquoMultiperiod mean-variance efficient portfolios with endogenous liabilitiesrdquoQuan-titative Finance vol 11 no 10 pp 1535ndash1546 2011

[8] M C Chiu and D Li ldquoAsset and liability management undera continuous-time mean-variance optimization frameworkrdquoInsurance vol 39 no 3 pp 330ndash355 2006

[9] S X Xie Z F Li and S Y Wang ldquoContinuous-time portfolioselection with liability mean-variancemodel and stochastic LQapproachrdquo Insurance vol 42 no 3 pp 943ndash953 2008


[10] S X Xie ldquoContinuous-time mean-variance portfolio selectionwith liability and regime switchingrdquo Insurance Mathematics ampEconomics vol 45 no 1 pp 148ndash155 2009

[11] Y Zeng and Z Li ldquoAsset-liability management under bench-mark and mean-variance criteria in a jump diffusion marketrdquoJournal of Systems Science amp Complexity vol 24 no 2 pp 317ndash327 2011

[12] R Strotz ldquoMyopia and inconsistency in dynamic utility maxi-mizationrdquoReview of Economic Studies vol 23 pp 165ndash180 1955

[13] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 pp 2970ndash30162010

[14] J Wang and P A Forsyth ldquoContinuous time mean varianceasset allocation a time-consistent strategyrdquo European Journal ofOperational Research vol 209 no 2 pp 184ndash201 2011

[15] X Y Cui D Li S Y Wang and S Zhu ldquoBetter than dynamicmean-variance time inconsistency and free cash flow streamrdquoMathematical Finance vol 22 no 2 pp 346ndash378 2012

[16] Z-P Chen G Li and J-E Guo ldquoOptimal investment policy inthe time consistent mean-variance formulationrdquo Insurance vol52 no 2 pp 145ndash156 2013

[17] C G Li Z F Li K Fu and H Q Song ldquoTime-consistentoptimal portfolio strategy for asset-liability management undermean-variance criterionrdquoAccounting and Finance Research vol2 pp 89ndash104 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

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Applied MathematicsJournal of


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Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



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Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


coincide with the strategy which computed at time 119905 + Δ119905As a result at time 119905 + Δ119905 the strategy computed at time 119905

will not be implemented by the investor unless there existssome commitment mechanism Strotz [12] first formalizedtime inconsistency and pointed out that the conflict couldbe solved by a time-consistent strategy Very recently muchmore scholars have paid their attentions on constructinga time-consistent mean-variance portfolio choice Amongothers Basak and Chabakauri [13] provided a fully analyt-ical characterization of the optimal time-consistent mean-variance portfolio within a general incomplete market econ-omy Wang and Forsyth [14] developed a numerical schemefor determining the optimal asset allocation strategy for time-consistent continuous time mean-variance optimization Byallowing the trade-off between the mean and the variance ofthe terminal wealth varying over time Cui et al [15] proposedaweak time consistency to comparewith Bellmanrsquos optimalityprinciple and derived an optimal mean-variance portfoliostrategy In all the literature mentioned above the studiesmainly referred to Bellmanrsquos optimality principle Howeverthe requirement (for short REQ) that local optimum isalso globally optimum is not necessarily satisfied whichis an essential requirement in solving the relevant optimalportfolio problem by the dynamic programming techniqueIn order to make up this shortfall Chen et al [16] by using atime consistent dynamic risk measure proposed a separabledynamic mean-variance model and showed that the relevantoptimal investment policy satisfies not only the Bellmanrsquosoptimality principle but also the REQ

Although asset-liability management is an importantissue in modern finance theory the time-consistent asset-liability management problem has not attracted enoughattention Recently Li et al [17] reported the time-consistentasset-liability management problem in the continuous-timesetting They employed Basak and Chabakaurirsquos [13] modelto study the continuous-time asset-liability managementproblem They derived the time-consistent optimal strategyand showed that the time-consistent efficient frontier withliability is below that without liability However the derivedtime-consistent policy does not satisfy the REQ As the afore-mentioned importance of the REQ in this paper we employthemodel of Chen et al [16] to analysis themultiperiod asset-liability management problem We derive time-consistentoptimal investment policies in a market with and withouta riskless asset respectively After comparing the optimaltime-consistent policies withmyopic strategies we show thatfor an arbitrarily risk averse investor if there is a risklessasset in the market the time-diversification effects arisingfrommultiperiod optimization can be ignored otherwise theeffects should be considered

This paper proceeds as follows In the next sectionwe formulate a time-consistent asset-liability managementmodel In Section 3 we derive the time-consistent optimalpolicy for a market without riskless asset Section 4 derivesthe time-consistent optimal policy for a market with bothriskless and risky assets Section 5 performs numericalexamples to illustrate our results The paper is concluded inSection 6

2 Model Formulation

Throughout this paper we assume that 119879 isin N is a fixed andfinite time horizon and trading only takes place at time 119896 =

0 1 119879 Let (ΩF 119901) be a probability space and let a 120590-fieldF

119905be the available information at time 119905

Consider a securitymarket consisting of one riskless assetand 119899 risky assets The return of the riskless asset at the 119905thinvestment period is assumed to be 119904

119905 The return of the 119894th

risky asset at the 119905th investment period is denoted by 119877119894

119905

and the relevant random return vector is denoted by 119877119905=

(1198771

119905 119877

119899

119905)1015840 which isF

119905+1-measurable

Consider an investor with an initial endowment 1199090and a

liability 1198970 We assume that the liability cannot be controlled

and denote by 119897119905the accumulative liability at time 119905 Let119876

119905be

the return of the liability at the 119905th investment period whichis F119905+1

-measurable It is clear that 119897119905+1

= 119876119905119897119905 Assume that

(1198771015840

119905 119876119905)1015840 119905 = 0 119879minus1 are statistically independentDenote

the expected return vector by 120583119905= 119864(119877

119905) and the variance-

covariancematrix by Σ119905= (Cov(119877119894

119905 119877119895

119905))119899times119899

which is assumedto be positive definite throughout this paper It is clear that119864(1198771199051198771015840

119905) = Σ119905+ 1205831199051205831015840

119905is also positive definite Denote by Σ

0

119905=

(Cov(119876119905 1198771

119905) Cov(119876

119905 119877119899

119905))1015840 the covariance vector The

investor begins hisher investment at time 0 and invests thecash amount119906119894

119905in the 119894th risky asset at the beginning of the 119905th

investment period where 119905 = 0 119879 minus 1 119906119905= (1199061

119905 119906

119899

119905)1015840

is called an investment decision at the 119905th investment periodand 119906 = (119906

0 119906

119879minus1) is an investment policy during the

entire investment horizonWe denote the wealth and the sur-plus of the investor at time 119905 by 119909

119905and 119878119905= 119909119905minus119897119905 respectively

A single-period conditional risk mapping is defined as

120588119905(119878119905+1

| F119905) = Var

119905(119878119905+1

) minus 120582119905119864119905(119878119905+1

) (1)

where 119864119905(119878119905+1

) = 119864(119878119905+1

| F119905) and Var

119905(119878119905+1

) = Var(119878119905+1

|

F119905) are the single-period conditional expectation of 119878

119905+1and

its conditional variance respectively Tradeoff parameter 120582119905

as a constant which is defined on [0infin] is a weight whichpresents the relative importance of expected profit comparedto the risk Note that if 120582

119905= 0 then the investor is arbitrary

risk averse who only focus on the risk if 120582119905= infin then the

investor is risk neutral who only concerns maximizing theirexpected profit if 120582

119905isin (0infin) the investor is risk averse who

considers both expected profit and the risk in his decisionIn multiperiod portfolio selection two main optimal

investment policies are the myopic and time-consistentstrategiesMyopic Strategy is a strategy whereby at each time119905 the investor determines their optimal investment decisionassuming the instantaneous moments of assets returns willremain fixed at their current values for the remainder of theinvestment horizon More specifically for any 119905 isin 0 119879 minus

1 the myopic strategy 119906119898119910

119905is a solution to the following

problem

min119906119905isinΠ119905

Var119905(119878119905+1

) minus 120582119905119864119905(119878119905+1

) (2)

where Π119905is a set of all permit policies at time 119905 Intuitively

a myopic investor only cares about the mean and variance ofthe surplus at the current period



119879

sum119895=119905+1

119864119905(120588119895minus1

(119878119895| F119895minus1

)) (3)


min119906isinΠ

119879

sum119895=1

119864 (120588119895minus1

(119878119895| F119895minus1

)) (4)


119895) | F119896) = 119864(sdot | F

119896) for 119895 gt 119896 we have

119879

sum119895=1

119864 (120588119895minus1

(119878119895| F119895minus1

))

= 1205880(1198781) + 119864 (120588

1(1198782)

+ 1198641(1205882(1198783) + sdot sdot sdot

+ 119864119879minus3

(120588119879minus2

(119878119879minus1

)

+119864119879minus2

(120588119879minus1

(119878119879))) ))


1205880(1198781)

+ 119864 (1205881(1198782) + 1198641(1205882(1198783)


(120588119879minus2

(119878119879minus1

)

+119864119879minus2

(120588119879minus1

(119878119879))) ))


min1199060

(1205880(1198781)

+ 119864min1199061

(1205881(1198782)

+ 1198641min1199062

(1205882(1198783) + sdot sdot sdot

+ 119864119879minus3

min119906119879minus2

(120588119879minus2

(119878119879minus1

)

+119864119879minus2

min119906119879minus1

(120588119879minus1

(119878119879))) )))

(7)







119909119905+1

= 1198771015840

119905119906119905 1198681015840119906119905= 119909119905 119905 = 0 1 119879 minus 1 (8)



min 1205880(1198781) + 1198640(1205881(1198782)

+ 1198641(1205882(1198783) + sdot sdot sdot

+ 119864119879minus3

(120588119879minus2

(119878119879minus1

)

+119864119879minus2

(120588119879minus1

(119878119879))) ))

st 1198771015840

119905119906119905= 119909119905+1

1198681015840119906119905= 119909119905

119897119905+1

= 119876119905119897119905

119878119905+1

= 119909119905+1

minus 119897119905+1

119905 = 0 1 119879 minus 1

(9)

Since Σ1




1198771015840

119879minus1)

that Σ1





119905gt 0



119906lowast

119905=

(Σ1

119905)minus1

119868

119886119905

119909119905+

1205821

119905

2(Σ1

119905)minus1

(120583119905minus 119902119905119868)

+ (Σ1

119905)minus1

(Σ0

119905minus 120574119905119868 + 120574119905+1

119864 (119876119905119877119905)) 119897119905

for 119905 = 0 119879 minus 1

(10)




119879minus1= 120582119879minus1

1205821

119905= 120582119905+ 1205821

119905+1119902119905+1

Σ1


Σ1

119905= Σ119905+ 119864 (119877

1199051198771015840

119905)119886119905+1

119886119879minus1

= 1198681015840(Σ1

119879minus1)minus1

119868 119886119905= 1198681015840(Σ1

119905)minus1

119868

119887119879minus1

= 1198681015840(Σ1

119879minus1)minus1

120583119879minus1

119887119905= 1198681015840(Σ1

119905)minus1

120583119905

119888119879minus1

= 1205831015840

119879minus1(Σ1

119879minus1)minus1

120583119879minus1

119888119905= 1205831015840

119905(Σ1

119905)minus1

120583119905

119889119879minus1

= 1198681015840(Σ1

119879minus1)minus1

Σ0

119879minus1119889119905= 1198681015840(Σ1

119905)minus1

Σ0

119905

119902119879minus1

= 119887119879minus1

119886119879minus1

119902119905= 119887119905119886119905

120572119879minus1

= 119888119879minus1


119887119879minus1

120572119905= 119888119905minus 119902119905119887119905

120574119879minus1

= 119889119879minus1

119886119879minus1

120574119905= 119889119905119886119905

120574119879minus1

= 120574119879minus1

120574119905= 120574119905+ 120574119905+1

119864(1198761199051198771015840

119905)(Σ1

119905)minus1

119868119886119905

1205731119879minus1

= 120582119879minus1

119864(119876119879minus1

) minus 120582119879minus1

(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1

1205731119905

= minus1205821

1199051205831015840

119905(Σ1

119905)minus1

(Σ0

119905minus 120574119905119868) minus 120582

1

119905120574119905+1

1205831015840

119905(Σ1

119905)minus1

119864(119876119905119877119905)

+ 120582119879minus1

119887119879minus1

120574119879minus1

+ (120582119905+ 1205731119905+1

)119864(119876119905)

1205732119879minus1

= Var(119876119879minus1

) minus (Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0

119879minus1+ 120574119879minus1

119889119879minus1

1205732119905

= [Var(119876119905) + 1205732119905+1

119864(1198762

119905)] + 119886

1199051205742

119905minus [Σ0

119905+ 120574119905+1

119864(119876119905119877119905)]1015840

sdot (Σ1

119905)minus1[Σ0

119905+ 120574119905+1

119864(119876119905119877119905)]




min Var119879minus1

(119878119879) minus 120582119879minus1

119864119879minus1

(119878119879)

st 1198771015840

119879minus1119906119879minus1

= 119909119879

1198681015840119906119879minus1

= 119909119879minus1

119897119879= 119876119879minus1

119897119879minus1

119878119879= 119909119879minus 119897119879

(11)


min 119881119879minus1

(119909119879minus1

119897119879minus1

)

= 1199061015840

119879minus1Σ1

119879minus1119906119879minus1

minus 2 [(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21199061015840

119879minus1] 119906119879minus1

+ Var (119876119879minus1

) 1198972

119879minus1+ 120582119879minus1

119864 (119876119879minus1

) 119897119879minus1

st 1198681015840119906119879minus1

= 119909119879minus1

(12)




119871 (119906119879minus1

) = 1199061015840

119879minus1Σ1

119879minus1119906119879minus1

minus 2 [(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] 119906119879minus1

+ Var (119876119879minus1

) 1198972

119879minus1+ 120582119879minus1

119864 (119876119879minus1

) 119897119879minus1

minus 120596119879(119909119879minus1

minus 1198681015840119906119879minus1

)

(13)


2Σ1

119879minus1119906119879minus1

minus 2(Σ0

119879minus1119897119879minus1

+120582119879minus1

2120583119879minus1

) + 120596119879119868 = 0 (14)

119909119879minus1

minus 1198681015840119906119879minus1

= 0 (15)


119906lowast

119879minus1= (Σ1

119879minus1)minus1

(Σ0

119879minus1119897119879minus1

+120582119879minus1

2120583119879minus1

) minus1

2120596119879(Σ1

119879minus1)minus1

119868

(16)

1198681015840119906lowast

119879minus1= 1198681015840(Σ1

119879minus1)minus1

(Σ0

119879minus1119897119879minus1

+120582119879minus1

2120583119879minus1

)

minus1

21205961198791198681015840(Σ1

119879minus1)minus1

119868

= 119889119879minus1

119897119879minus1

+120582119879minus1

2119887119879minus1

minus1

2119886119879minus1

120596119879

= 119909119879minus1

(17)


120596119879= 2

119889119879minus1

119886119879minus1

119897119879minus1

+ 120582119879minus1

119887119879minus1

119886119879minus1

minus 21

119886119879minus1

119909119879minus1

= 2120574119879minus1

119897119879minus1

+ 120582119879minus1

119902119879minus1

minus 21

119886119879minus1

119909119879minus1

(18)


119906lowast

119879minus1=

(Σ1

119879minus1)minus1

119868

119886119879minus1

119909119879minus1

+ (Σ1

119879minus1)minus1

[Σ0


119868] 119897119879minus1

+120582119879minus1

2(Σ1

119879minus1)minus1

[120583119879minus1


119868]

(19)





119881lowast

119879minus1(119909119879minus1

119897119879minus1

) =1

119886119879minus1

1199092


119902119879minus1

119909119879minus1


119909119879minus1

119897119879minus1

+ 1205731119879minus1

119897119879minus1

+ 1205732119879minus1

1198972

119879minus1minus

(120582119879minus1

)2

4120572119879minus1

(20)




min Var119879minus2

(119878119879minus1

) minus 120582119879minus2

119864119879minus2

(119878119879minus1

)

+ 119864119879minus2

(119881lowast

119879minus1(119909119879minus1

119897119879minus1

))

st 1198771015840

119879minus2119906119879minus2

= 119909119879minus1

1198681015840119906119879minus2

= 119909119879minus2

119897119879minus1

= 119876119879minus2

119897119879minus2

119878119879minus1

= 119909119879minus1


(21)


119864119879minus2

(119881lowast

119879minus1(119909119879minus1

119897119879minus1

))

=1

119886119879minus1

119864119879minus2

(1199092


119902119879minus1

119864119879minus2

(119909119879minus1

)


119864119879minus2

(119909119879minus1

119897119879minus1

) + 1205731119879minus1

119864119879minus2

(119897119879minus1

)

+ 1205732119879minus1

119864119879minus2

(1198972

119879minus1) minus

(120582119879minus1

)2

4120572119879minus1

(22)


(119881lowast

119879minus1(119909119879minus1

119897119879minus1


min 119881119879minus2

(119909119879minus2

119897119879minus2

)

st 1198681015840119906119879minus2

= 119909119879minus2

(23)

where

119881119879minus2

(119909119879minus2

119897119879minus2

) = 1199061015840

119879minus2Σ1

119879minus2119906119879minus2

minus 2 [(Σ0

119879minus2)1015840

+ 120574119879minus1

119864 (119876119879minus2

1198771015840

119879minus2)] 119897119879minus2

+1205821

119879minus2

21205831015840

119879minus2119906119879minus2

+ [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1


)2

4120572119879minus1

(24)



119871 (119906119879minus2

) = 1199061015840

119879minus2Σ1

119879minus2119906119879minus2

minus 2Φ1015840

119879minus2119906119879minus2

+ Ψ119879minus2


(119909119879minus2

minus 1198681015840119906119879minus2

)

(25)

where

Φ119879minus2

= [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)] 119897119879minus2

+1205821

119879minus2

2120583119879minus2

Ψ119879minus2

= [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1


)2

4120572119879minus1

(26)


2Σ1

119879minus2119906119879minus2


+ 120596119879minus1

119868 = 0 (27)

119909119879minus2

minus 1198681015840119906119879minus2

= 0 (28)

From (27) we have

119906lowast

119879minus2= (Σ1

119879minus2)minus1

Φ119879minus2

minus1

2120596119879minus1

(Σ1

119879minus2)minus1

119868 (29)

1198681015840119906lowast

119879minus2= 1198681015840(Σ1

119879minus2)minus1

Φ119879minus2

minus1

2120596119879minus1

1198681015840(Σ1

119879minus2)minus1

119868

= 1198681015840(Σ1

119879minus2)minus1

Φ119879minus2

minus1

2120596119879minus1

119886119879minus2

= 119909119879minus2

(30)

which implies

120596119879minus1

= 21198681015840(Σ1

119879minus2)minus1

Φ119879minus2

119886119879minus2

minus 21

119886119879minus2

119909119879minus2

(31)


into (29) we get

119906lowast

119879minus2=

(Σ1

119879minus2)minus1

119868

119886119879minus2

119909119879minus2

+ (Σ1

119879minus2)minus1

Φ119879minus2

minus1198681015840(Σ1

119879minus2)minus1

Φ119879minus2

119886119879minus2

(Σ1

119879minus2)minus1

119868

(32)



119906lowast

119879minus2=

(Σ1

119879minus2)minus1

119868

119886119879minus2

119909119879minus2

+1205821

119879minus2

2(Σ1

119879minus2)minus1

(120583119879minus2


119868)

+ (Σ1

119879minus2)minus1

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868] 119897119879minus2

(33)




details)

119881lowast

119879minus2(119909119879minus2

119897119879minus2

) =1

119886119879minus2

1199092


119879minus2119902119879minus2

119909119879minus2


119909119879minus2

119897119879minus2

+ 1205731119879minus2

119897119879minus2

+ 1205732119879minus2

1198972

119879minus2


)2

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(34)


119881lowast

119905(119909119905 119897119905) =

1

119886119905

1199092

119905minus 1205821

119905119902119905119909119905minus 2120574119905119909119905119897119905

+ 1205731119905119897119905+ 12057321199051198972

119905minus

119879minus1

sum119894=119905

(1205821

119894)2

4120572119894

(35)


119905minus1and



min Var119905minus1

(119878119905) minus 120582119905minus1

119864119905minus1

(119878119905)

+ 119864119905minus1

(119881lowast

119905(119909119905 119897119905))

st 1198771015840

119905minus1119906119905minus1

= 119909119905

1198681015840119906119905minus1

= 119909119905minus1

119897119905= 119876119905minus1

119897119905minus1

119878119905= 119909119905minus 119897119905

(36)


119864119905minus1

(119881lowast

119905(119909119905 119897119905)) =

1

119886119905

119864119905minus1

(1199092

119905) minus 1205821

119905119902119905119864119905minus1

(119909119905)

minus 2120574119905119864119905minus1

(119909119905119897119905) + 1205731119905119864119905minus1

(119897119905)

+ 1205732119905119864119905minus1

(1198972

119905) minus

119879minus1

sum119894=119905

(1205821

119894)2

4120572119894

(37)


(119881lowast



min 119881119905minus1

(119909119905minus1

119897119905minus1

) = 1199061015840

119879minus1Σ1

119905minus1119906119905minus1

minus 2Φ1015840

119905minus1119906119905minus1

+ Ψ119905minus1

st 1198681015840119906119905minus1

= 119909119905minus1

(38)

where

Φ119905minus1

= [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] 119897119905minus1

+1205821

119905minus1

2120583119905minus1

Ψ119905minus1

= [Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] 1198972

119905minus1

+ [120582119905minus1

+ 1205731119905] 119864 (119876

119905minus1) 119897119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4120572119894

(39)



119871 (119906119905minus1

) = 1199061015840

119879minus1Σ1

119905minus1119906119905minus1

minus 2Φ1015840

119905minus1119906119905minus1

+ Ψ119905minus1

minus 120596119905(119909119905minus1

minus 1198681015840119906119905minus1

)

(40)


2Σ1

119905minus1119906119905minus1


+ 120596119905119868 = 0

119909119905minus1

minus 1198681015840119906119905minus1

= 0

(41)

Thus we have

119906lowast

119905minus1= (Σ1

119905minus1)minus1

Φ119905minus1

minus1

2120596119905(Σ1

119905minus1)minus1

119868 (42)

1198681015840119906lowast

119905minus1= 1198681015840(Σ1

119905minus1)minus1

Φ119905minus1

minus1

21205961199051198681015840(Σ1

119905minus1)minus1

119868

= 1198681015840(Σ1

119905minus1)minus1

Φ119905minus1

minus1

2120596119905119886119905minus1

= 119909119905minus1

(43)

which implies

120596119905= 2

1198681015840(Σ1

119905minus1)minus1

Φ119905minus1

119886119905minus1

minus 21

119886119905minus1

119909119905minus1

(44)


119906lowast

119905minus1=

(Σ1

119905minus1)minus1

119868

119886119905minus1

119909119905minus1

minus1198681015840(Σ1

119905minus1)minus1

Φ119905minus1

119886119905minus1

(Σ1

119905minus1)minus1

119868

+ (Σ1

119905minus1)minus1

Φ119905minus1

=(Σ1

119905minus1)minus1

119868

119886119905minus1

119909119905minus1

+1205821

119905minus1

2(Σ1

119905minus1)minus1

(120583119905minus1


119868)

+ (Σ1

119905minus1)minus1

[Σ0


119868 + 120574119905119864 (119876119905minus1

119877119905minus1

)] 119897119905minus1

(45)




119881lowast

119905minus1(119909119905minus1

119897119905minus1

) =1

119886119905minus1

1199092


119905minus1119902119905minus1

119909119905minus1


119909119905minus1

119897119905minus1

+ 1205731119905minus1

119897119905minus1

+ 1205732119905minus1

1198972

119905minus1minus

119879minus1

sum119894=119905minus1

(1205821

119894)2

4120572119894





119906lowast

119905=

(Σ1

119905)minus1

119868

119886119905

119909119905+

1205821

119905

2(Σ1

119905)minus1

(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1

(47)





119906lowast

119905=

(Σ1

119905)minus1

119868

119886119905

119909119905+

1205821

119905

2(Σ1

119905)minus1

(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1

(48)


119905= 1198681015840(Σ1

119905)minus1Σ0

119905=

0 120574119905= 119889119905119886119905= 0 and 120574


119906lowast

119905=

(Σ1

119905)minus1

119868

119886119905

119909119905+

1205821

119905

2(Σ1

119905)minus1

(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1

(49)




(Σ1

119905)minus1

(Σ0

119905minus 120574119905119868 + 120574119905+1

119864 (119876119905119877119905)) 119897119905 (50)



119905


min119906119905

Var119905(119878119905+1

) minus 120582119905119864119905(119878119905+1

)

st 1198771015840

119905119906119905= 119909119905+1

1198681015840119906119905= 119909119905

119897119905+1

= 119876119905119897119905

119878119905+1

= 119909119905+1

minus 119897119905+1

(51)


119906119898119910

119905=

(Σ119905)minus1

119868

119886119905

119909119905+

120582119905

2(Σ119905)minus1

(120583119905minus 119902119905119868)

+ (Σ119905)minus1

(Σ0

119905minus 120574119905119868) 119897119905 119905 = 0 119879 minus 1

(52)


119905119886119905


119905rarr 0


119906lowast

119905=

(Σ1

119905)minus1

119868

119886119905

119909119905+ (Σ1

119905)minus1

(Σ0

119905minus 120574119905119868 + 120574119905+1

119864 (119876119905119877119905)) 119897119905

119905 = 0 119879 minus 1

119906119898119910

119905=

(Σ119905)minus1

119868

119886119905

119909119905+ (Σ119905)minus1

(Σ0

119905minus 120574119905119868) 119897119905 119905 = 0 119879 minus 1

(53)







119909119905= 1198751015840

119905minus1119906119905minus1

+ 119904119905minus1

119909119905minus1

(54)



0(1198781) + 1198640(1205881(1198782) + 1198641(1205882(1198783) + sdot sdot sdot

+119864119879minus3

(120588119879minus2

(119878119879minus1

) + 119864119879minus2

(120588119879minus1

(119878119879))) ))

st 119909119905+1

= 1198751015840

119905119906119905+ 119904119905119909119905

119897119905+1

= 119876119905119897119905

119878119905+1

= 119909119905+1

minus 119897119905+1

119905 = 0 1 119879 minus 1

(55)



119906lowast

119905= Σminus1

119905120593119905119897119905+

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (56)

where 120593119905= 119864(119875

119905119876119905) minus 119864(119875

119905)119864(119876119905) and 120598

119905= 119864(119875

119905)




119879minus1= 120582119879minus1

1205821

119905= 120582119905+ 1205821

119905+1119904119905+1

1205731119879minus1

= 120582119879minus1

[119864 (119876119879minus1

) minus 1205981015840


)minus1

120593119879minus1

] 1205731119905

= (120582119905+ 1205731119905+1

) 119864(119876119905) minus 1205821

1199051205981015840

119905(Σ119905)minus1120593119905

1205732119879minus1

= Var(119876119879minus1

) minus 1205931015840


)minus1

120593119879minus1

1205732119905

= Var(119876119905) + 1205732119905+1

119864(1198762

119905) minus 1205931015840

119905(Σ119905)minus1120593119905




min119906119879minus1

Var119879minus1

(119878119879) minus 120582119879minus1

119864119879minus1

(119878119879)

st 119909119879

= 119904119879minus1

119909119879minus1

+ 1198751015840

119879minus1119906119879minus1

119897119879

= 119876119879minus1

119897119879minus1

119878119879= 119909119879minus 119897119879

(57)


min119906119879minus1

119881119879minus1

(119909119879minus1

119897119879minus1

) (58)

where

119881119879minus1

(119909119879minus1

119897119879minus1

) = 1199061015840


119906119879minus1

minus 2(1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840

119879minus1)119906119879minus1

+ Var (119876119879minus1

) 1198972


119904119879minus1

119909119879minus1

+ 120582119879minus1

119864 (119876119879minus1

) 119897119879minus1

(59)


119889 (119881119879minus1

(119909119879minus1

119897119879minus1

))

119889 (119906119879minus1

)= 2Σ119879minus1

119906119879minus1

minus 2(120593119879minus1

119897119879minus1

+120582119879minus1

2120598119879minus1

)

= 0

(60)

which implies

119906lowast

119879minus1= (Σ119879minus1

)minus1

[120593119879minus1

119897119879minus1

+120582119879minus1

2120598119879minus1

] (61)




119881lowast

119879minus1(119909119879minus1

119897119879minus1

) = 1205732119879minus1

1198972

119879minus1+ 1205731119879minus1

119897119879minus1


119904119879minus1

119909119879minus1


)2

4119872119879minus1

(62)


= 1205981015840


119879minus1120598119879minus1




min Var119879minus2

(119878119879minus1

) minus 120582119879minus2

119864119879minus2

(119878119879minus1

)

+ 119864119879minus2

(119881lowast

119879minus1(119909119879minus1

119897119879minus1

))

st 119909119879minus1

= 119904119879minus2

119909119879minus2

+ 1198751015840

119879minus2119906119879minus2

119897119879minus1

= 119876119879minus2

119897119879minus2

119878119879minus1

= 119909119879minus1


(63)


119864119879minus2

(119881lowast

119879minus1(119909119879minus1

119897119879minus1

)) = 1205732119879minus1

119864 (1198762

119879minus2) 1198972

119879minus2

+ 1205731119879minus1

119864 (119876119879minus2

) 119897119879minus2


119904119879minus1

119904119879minus2

119909119879minus2


119904119879minus1

1205981015840

119879minus2119906119879minus2


)2

4119872119879minus1

(64)


(119881lowast

119879minus1(119909119879minus1

119897119879minus1


min119906119879minus2

119881119879minus2

(119909119879minus2

119897119879minus2

) (65)


(119909119879minus2

119897119879minus2

) = 1199061015840


119906119879minus2

minus 2(1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840

119879minus2)119906119879minus2

+ [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

(66)


119889 (119881119879minus2

(119909119879minus2

119897119879minus2

))

119889 (119906119879minus2

)= 2Σ119879minus2

119906119879minus2

minus 2(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= 0

(67)


Thus

119906lowast

119879minus2= (Σ119879minus2

)minus1

(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= (Σ119879minus2

)minus1

120593119879minus2

119897119879minus2

+1205821

119879minus2

2(Σ119879minus2

)minus1

120598119879minus2

(68)




119881lowast

119879minus2(119909119879minus2

119897119879minus2

) = 1205732119879minus2

1198972

119879minus2+ 1205731119879minus2

119897119879minus2

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

minus(1205821

119879minus2)2

4119872119879minus2

(69)


= 1205981015840


119879minus2120598119879minus2


both (56) and

119881lowast

119905(119909119905 119897119905) = 12057321199051198972

119905+ 1205731119905119897119905minus 1205821

119905119904119905119909119905minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

(70)

hold where 119872119894= 1205981015840

119894Σminus1



119905minus1and



min Var119905minus1

(119878119905) minus 120582119905minus1

119864119905minus1

(119878119905) + 119864119905minus1

(119881lowast

119905(119909119905 119897119905))

st 119909119905= 119904119905minus1

119909119905minus1

+ 1198751015840

119905minus1119906119905minus1

119897119905= 119876119905minus1

119897119905minus1

119878119905= 119909119905minus 119897119905

(71)


119864119905minus1

(119881lowast

119905(119909119905 119897119905)) = 120573

2119905119864 (1198762

119905minus1) 1198972

119905minus1+ 1205731119905119864 (119876119905minus1

) 119897119905

minus 1205821

119905119904119905119904119905minus1

119909119905minus1

minus 1205821

1199051199041199051205981015840

119905minus1119906119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

(72)


(119881lowast



119881119905minus1

(119909119905minus1

119897119905minus1

) (73)

where

119881119905minus1

(119909119905minus1

119897119905minus1

) = 1199061015840


119906119905minus1

minus 2(1205931015840

119905minus1119897119905minus1

+1205821

119905minus1

21205981015840

119905minus1)119906119905minus1

+ [Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] 1198972

119905minus1

+ [120582119905minus1

119864 (119876119905minus1

) + 1205731119905119864 (119876119905minus1

)] 119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

(74)


119889 (119881119905minus1

(119909119905minus1

119897119905minus1

))

119889 (119906119905minus1

)= 2Σ119905minus1

119906119905minus1

minus 2(120593119905minus1

119897119905minus1

+1205821

119905minus1

2120598119905minus1

)

= 0

(75)

which implies

119906lowast


119905minus1120593119905minus1

119897119905minus1

+1205821

119905minus1

2Σminus1

119905minus1120598119905minus1

(76)




119881lowast

119905minus1(119909119905minus1

119897119905minus1

) = 1205732119905minus1

1198972

119905minus1+ 1205731119905minus1

119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905minus1

(1205821

119894)2

4119872119894

(77)







119906lowast

119905=

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (78)






119906lowast

119905=

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (79)


120593119905= 119864 (119875

119905119876119905) minus 119864 (119875

119905) 119864 (119876

119905)

= 119864 ((119877119905minus 119904119905119868)119876119905) minus 119864 (119877

119905minus 119904119905119868) 119864 (119876

119905)

= 119864 (119877119905119876119905) minus 119864 (119877

119905) 119864 (119876

119905)

= Σ0

119905

= 0

(80)



119906lowast

119905=

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (81)




Σminus1

119905120593119905119897119905 (82)



min119906119905

Var119905(119878119905+1

) minus 120582119905119864119905(119878119905+1

)

st 119909119905+1

= 119904119905119909119905+ 1198751015840

119905119906119905

119897119905+1

= 119876119905119897119905

119878119905+1

= 119909119905+1

minus 119897119905+1

(83)


119906119898119910

119905= Σminus1

119905120593119905119897119905+

120582119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (84)




119906lowast

119905= Σminus1

119905120593119905119897119905 119905 = 0 119879 minus 1 (85)






119905


lowast

119905

0 (

11941198970+ 0903119909

0+ 3736

01071198970minus 0066119909

0+ 0203

minus05771198970+ 0163119909

0minus 3939

) (

09031199090+ 3736

minus00661199090+ 0203

01631199090minus 3939

)

1 (

minus03721198971+ 1018119909

1+ 1743

01981198971minus 0060119909

1+ 0095

minus06331198971+ 0042119909

1minus 1838

) (

10181199091+ 1743

minus00601199091+ 0095

00421199091minus 1838

)




my119905

0 (

09031199090+ 1194119897

0+ 3736

minus00661199090+ 0107119897

0+ 0203

01631199090minus 0577119897

0minus 3939

) (

10181199090+ 0449119897

0+ 1743

minus00601199090+ 0150119897

0+ 0095

00421199090minus 0599119897

0minus 1838

)

1 (

10181199091minus 0372119897

1+ 1743

minus00601199091+ 0198119897

1+ 0095

00421199091minus 0633119897

1minus 1838

) (

10181199091+ 0449119897

1+ 1743

minus00601199091+ 0150119897

1+ 0095

00421199091minus 0599119897

1minus 1838

)



119905=

(1162 1246 1228) and

Σ119905= (

00146 00187 00145

00187 00854 00104

00145 00104 00289

) (86)






119905=

05 and 120582119905







my119905

0 (

09031199090+ 1194119897

0

minus00661199090+ 0107119897

0

01631199090minus 0577119897

0

) (

10181199090+ 0449119897

0

minus00601199090+ 0150119897

0

00421199090minus 0599119897

0

)

1 (

10181199091minus 0372119897

1

minus00601199091+ 0198119897

1

00421199091minus 0633119897

1

) (

10181199091+ 0449119897

1

minus00601199091+ 0150119897

1

00421199091minus 0599119897

1

)




my119905

0 (

04441198970+ 7889

01501198970minus 0062

minus05991198970minus 3572

) (

04441198970+ 3867

01501198970minus 0031

minus05991198970minus 1751

)

1 (

04441198971+ 3867

01501198971minus 0031

minus05991198971minus 1751

) (

04441198971+ 3867

01501198971minus 0031

minus05991198971minus 1751

)




my119905

0 (

04441198970

01501198970

minus05991198970

) (

04441198970

01501198970

minus05991198970

)

1 (

04441198971

01501198971

minus05991198971

) (

04441198971

01501198971

minus05991198971

)


119905= 120583119905minus 119904119905119868 =

(0122 0206 0188)1015840 and120593

119905= Σ0

119905= (00006 00149 00050)





6 Conclusion


Appendices

A Proof of (20)


119879minus1into 119906

lowast

119879minus1

1015840Σ1


119879minus1gives

119906lowast

119879minus1

1015840

Σ1


119879minus1=

119868

119886119879minus1

119909119879minus1

+ [Σ0


119868] 119897119879minus1

+120582119879minus1

2[120583119879minus1


119868]

1015840

(Σ1

119879minus1)minus1

times 119868

119886119879minus1

119909119879minus1

+ [Σ0


119868]


+120582119879minus1

2[120583119879minus1


119868]

=1

119886119879minus1

1199092

119879minus1+ [Σ0


119868]1015840

times (Σ1

119879minus1)minus1

[Σ0


119868] 1198972

119879minus1

+ 120582119879minus1

[Σ0


119868]1015840

times (Σ1

119879minus1)minus1

[120583119879minus1


119868] 119897119879minus1

+(120582119879minus1

)2

4[120583119879minus1


119868]1015840

times (Σ1

119879minus1)minus1

[120583119879minus1


119868]

=1

119886119879minus1

1199092

119879minus1+ [(Σ

0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0

119879minus1


1205742

119879minus1] 1198972

119879minus1

+ [(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1


120574119879minus1

]


119897119879minus1

+(120582119879minus1

)2

4120572119879minus1

(A1)



0

119879minus1)1015840

119897119879minus1

+

(120582119879minus1

2)1205831015840



2 [(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840


119879minus1

=2

119886119879minus1

[(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1

119868119909119879minus1

+ 2 [(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1


times [Σ0


119868] 119897119879minus1

+ 120582119879minus1

[(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1



119868]

= 2120574119879minus1

119897119879minus1

119909119879minus1

+ 119902119879minus1

120582119879minus1

119909119879minus1

+ 2 [(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0


119889119879minus1

] 1198972

119879minus1

+ 2120582119879minus1

[(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1


120574119879minus1

] 119897119879minus1

+(120582119879minus1

)2

2120572119879minus1

(A2)

Thus119881lowast

119879minus1(119909119879minus1

119897119879minus1

)

=1

119886119879minus1

1199092


119902119879minus1

119909119879minus1


119909119879minus1

119897119879minus1

+ 120582119879minus1

[119864 (119876119879minus1

) minus (Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1

+119887119879minus1

120574119879minus1

] 119897119879minus1

+ [Var (119876119879minus1

) minus (Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0

119879minus1

+120574119879minus1

119889119879minus1

] 1198972

119879minus1minus

(120582119879minus1

)2

4120572119879minus1

=1

119886119879minus1

1199092


119902119879minus1

119909119879minus1


119909119879minus1

119897119879minus1

+ 1205731119879minus1

119897119879minus1

+ 1205732119879minus1

1198972

119879minus1minus

(120582119879minus1

)2

4120572119879minus1

(A3)


1198681015840

119886119879minus2

(Σ1

119879minus2)minus1

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

=119889119879minus2

119886119879minus2

+ 120574119879minus1

1198681015840(Σ1

119879minus2)minus1

119864 (119876119879minus2

119877119879minus2

)

119886119879minus2


= 120574119879minus2

+ 120574119879minus1

1198681015840(Σ1

119879minus2)minus1

119864 (119876119879minus2

119877119879minus2

)

119886119879minus2


= 120574119879minus2


= 0

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1

[120583119879minus2


119868] =119887119879minus2

119886119879minus2


= 0

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

119868

= 119886119879minus2

120574119879minus2

+ 120574119879minus1

119864 (119876119879minus2

1198771015840

119879minus2) (Σ1

119879minus2)minus1

119868

119886119879minus2

= 119886119879minus2

120574119879minus2

(B1)


119879minus2

1015840Σ1




and 119897119879minus2

119909119879minus2

in 119906lowast

119879minus2

1015840Σ1



119879minus2is

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1 119868

119886119879minus2

=1

119886119879minus2

(B2)


is

1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

= 1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

minus 1205821

119879minus2120574119879minus2

1198681015840(Σ1

119879minus2)minus1

[120583119879minus2


119868]

= 1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

(B3)


is

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]1015840

sdot (Σ1

119879minus2)minus1

sdot [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

= [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]


1205742

119879minus2

(B4)


(1205821

119879minus2)2

4[120583119879minus2


119868]1015840

(Σ1

119879minus2)minus1

[120583119879minus2


119868]

=(1205821

119879minus2)2

4120572119879minus2

(B5)




119879minus2


is

1205821

119879minus2

2

1205831015840

119879minus2(Σ1

119879minus2)minus1

119868

119886119879minus2

=1205821

119879minus2119902119879minus2

2 (B6)



119909119879minus2

is 120574119879minus2


is

1205821

119879minus2

2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

(120583119879minus2


119868)

+1205821

119879minus2

21205831015840

119879minus2(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

(B7)


is

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

= [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)] minus 119886119879minus2

1205742

119879minus2

(B8)


(1205821

119879minus2

2)

2

1205831015840

119879minus2(Σ1

119879minus2)minus1

(120583119879minus2


119868) = (1205821

119879minus2

2)

2

120572119879minus2

(B9)


119879minus2(119909119879minus2

119897119879minus2


119879minus2is

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1 119868

119886119879minus2

=1

119886119879minus2

(B10)



119902119879minus2


119909119879minus2



is

[Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] + 119886

119879minus21205742

119879minus2

minus [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]

= 1205732119879minus2

(B11)


is

minus 1205821

119879minus21205831015840

119879minus2(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)]

= 1205731119879minus2

(B12)


minus1205822

119879minus1

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(B13)

Thus

119881lowast

119879minus2(119909119879minus2

119897119879minus2

) =1

119886119879minus2

1199092


119879minus2119902119879minus2

119909119879minus2


119909119879minus2

119897119879minus2

+ 1205731119879minus2

119897119879minus2

+ 1205732119879minus2

1198972

119879minus2minus

1205822

119879minus1

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(B14)


1198681015840

119886119905minus1

(Σ1

119905minus1)minus1

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

=119889119905minus1

119886119905minus1

+ 120574119905

1198681015840(Σ1

119905minus1)minus1

119864 (119876119905minus1

119877119905minus1

)

119886119905minus1


= 120574119905minus1

+ 120574119905

1198681015840(Σ1

119905minus1)minus1

119864 (119876119905minus1

119877119905minus1

)

119886119905minus1


= 120574119905minus1


= 0

1198681015840

119886119905minus1

(Σ1

119905minus1)minus1

[120583119905minus1


119868] =119887119905minus1

119886119905minus1


= 0

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

119868

= 119886119905minus1

120574119905minus1

+ 120574119905

119864 (119876119905minus1

1198771015840

119905minus1) (Σ1

119905minus1)minus1

119868

119886119905minus1

= 119886119905minus1

120574119905minus1

(C1)


119905minus1

1015840Σ1




and 119897119905minus1

119909119905minus1


is

1198681015840

119886119905minus1

(Σ1

119905minus1)minus1 119868

119886119905minus1

=1

119886119905minus1

(C2)


is

1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

= 1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

minus 1205821

119905minus1120574119905minus1

1198681015840(Σ1

119905minus1)minus1

[120583119905minus1


119868]


= 1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

(C3)


is

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]1015840

sdot (Σ1

119905minus1)minus1

sdot [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

= [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] minus 119886119905minus1

1205742

119905minus1

(C4)


(1205821

119905minus1)2

4[120583119905minus1


119868]1015840

(Σ1

119905minus1)minus1

[120583119905minus1


119868] =(1205821

119905minus1)2

4120572119905minus1

(C5)




119905minus1is


is

1205821

119905minus1

2

1205831015840

119905minus1(Σ1

119905minus1)minus1

119868

119886119905minus1

=1205821

119905minus1119902119905minus1

2 (C6)


119909119905minus1

is 120574119905minus1


is

1205821

119905minus1

2[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1



119868) +1205821

119905minus1

21205831015840

119905minus1(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

(C7)


is

[Σ0

119905minus1+ 120574119905119864(119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

= [Σ0

119905minus1+ 120574119905119864(119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] minus 119886119905minus1

1205742

119905minus1

(C8)


(1205821

119905minus1

2)

2

120583119905minus1

(Σ1

119905minus1)minus1

(120583119905minus1


119868) = (1205821

119905minus1

2)

2

120572119905minus1

(C9)


119905minus1(119909119905minus1

119897119905minus1


119905minus1is 1119886



1

119905minus1119902119905minus1


119905minus1119909119905minus1



is

[Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] + 119886

119905minus11205742

119905minus1

minus [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]

= 1205732119905minus1

(C10)


is

minus 1205821

119905minus11205831015840

119905minus1(Σ1

119905minus1)minus1

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

+ (120582119905minus1

+ 1205731119905) 119864 (119876

119905minus1) = 1205731119905minus1

(C11)


(1205821

119894)2120572119894

Thus

119881lowast

119905minus1(119909119905minus1

119897119905minus1

) =1

119886119905minus1

1199092


119905minus1119902119905minus1

119909119905minus1


119909119905minus1

119897119905minus1

+ 1205731119905minus1

119897119905minus1

+ 1205732119905minus1

1198972

119905minus1minus

1

4

119879minus1

sum119894=119905minus1

(1205821

119894)2

120572119894

(C12)

D Proof of (62)


119879minus1into 119906

lowast

119879minus1

1015840Σ119879minus1

119906lowast


119906lowast

119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1= (1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840

119879minus1)


)minus1

(120593119879minus1

119897119879minus1

+120582119879minus1

2120598119879minus1

)

= 1205931015840


)minus1

120593119879minus1

1198972

119879minus1+ 120582119879minus1

1205981015840

119879minus1


)minus1

120593119879minus1

119897119879minus1

+(120582119879minus1

)2

4119872119879minus1

2 (1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840



119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1

(D1)

Thus

119881lowast

119879minus1(119909119879minus1

119897119879minus1


119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1+ Var (119876

119879minus1) 1198972

119879minus1


119904119879minus1

119909119879minus1

+ 120582119879minus1

119864 (119876119879minus1

) 119897119879minus1

= [Var (119876119879minus1

) minus 1205931015840


)minus1

120593119879minus1

] 1198972

119879minus1

+ 120582119879minus1

[119864 (119876119879minus1

) minus 1205981015840


)minus1

120593119879minus1

]



119904119879minus1

119909119879minus1


)2

4119872119879minus1

= 1205732119879minus1

1198972

119879minus1+ 1205731119879minus1

119897119879minus1


119904119879minus1

119909119879minus1


)2

4119872119879minus1

(D2)



lowast


119906lowast

119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2= (1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840

119879minus2)


)minus1

(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= 1205931015840


)minus1

120593119879minus2

1198972

119879minus2+ 1205821

119879minus21205981015840

119879minus2


)minus1

120593119879minus2

119897119879minus2

+(1205821

119879minus2)2

4119872119879minus2

2 1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840



119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2

(E1)Thus

119881lowast

119879minus2(119909119879minus2

119897119879minus2


119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2

+ [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

= 1205732119879minus2

1198972

119879minus2+ 1205731119879minus2

119897119879minus2

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

minus(1205821

119879minus2)2

4119872119879minus2

(E2)where1205731119879minus2

= (120582119879minus2

+ 1205731119879minus1

) 119864 (119876119879minus2

) minus 1205821

119879minus21205981015840


)minus1

120593119879minus2

1205732119879minus2

= Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2) minus 1205931015840


)minus1

120593119879minus2

(E3)


lowast

119905minus1that

119906lowast

119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1= (1205931015840

119879minus1119897119905minus1

+1205821

119905minus1

21205981015840

119879minus1)


)minus1

(120593119905minus1

119897119905minus1

+1205821

119905minus1

2120598119905minus1

)

= 1205931015840


)minus1

120593119905minus1

1198972

119905minus1+ 1205821

119905minus11205981015840

119879minus1


)minus1

120593119905minus1

119897119905minus1

+(1205821

119905minus1)2

4119872119905minus1

2 (1205931015840

119879minus1119897119905minus1

+1205821

119905minus1

21205981015840



119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1

(F1)

Thus

119881lowast

119905minus1(119909119905minus1

119897119905minus1


119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1

+ [Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] 1198972

119905minus1

+ [120582119905minus1

119864 (119876119905minus1

) + 1205731119905119864 (119876119905minus1

)] 119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

= 1205732119905minus1

1198972

119905minus1+ 1205731119905minus1

119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905minus1

(1205821

119894)2

4119872119894

(F2)

where

1205731119905minus1

= (120582119905minus1

+ 1205731119905) 119864 (119876

119905minus1) minus 1205821

119905minus11205981015840


)minus1

120593119905minus1

1205732119905minus1

= Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1) minus 1205931015840


)minus1

120593119905minus1

(F3)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119879

sum119895=119905+1

119864119905(120588119895minus1

(119878119895| F119895minus1

)) (3)


min119906isinΠ

119879

sum119895=1

119864 (120588119895minus1

(119878119895| F119895minus1

)) (4)


119895) | F119896) = 119864(sdot | F

119896) for 119895 gt 119896 we have

119879

sum119895=1

119864 (120588119895minus1

(119878119895| F119895minus1

))

= 1205880(1198781) + 119864 (120588

1(1198782)

+ 1198641(1205882(1198783) + sdot sdot sdot

+ 119864119879minus3

(120588119879minus2

(119878119879minus1

)

+119864119879minus2

(120588119879minus1

(119878119879))) ))


1205880(1198781)

+ 119864 (1205881(1198782) + 1198641(1205882(1198783)


(120588119879minus2

(119878119879minus1

)

+119864119879minus2

(120588119879minus1

(119878119879))) ))


min1199060

(1205880(1198781)

+ 119864min1199061

(1205881(1198782)

+ 1198641min1199062

(1205882(1198783) + sdot sdot sdot

+ 119864119879minus3

min119906119879minus2

(120588119879minus2

(119878119879minus1

)

+119864119879minus2

min119906119879minus1

(120588119879minus1

(119878119879))) )))

(7)







119909119905+1

= 1198771015840

119905119906119905 1198681015840119906119905= 119909119905 119905 = 0 1 119879 minus 1 (8)



min 1205880(1198781) + 1198640(1205881(1198782)

+ 1198641(1205882(1198783) + sdot sdot sdot

+ 119864119879minus3

(120588119879minus2

(119878119879minus1

)

+119864119879minus2

(120588119879minus1

(119878119879))) ))

st 1198771015840

119905119906119905= 119909119905+1

1198681015840119906119905= 119909119905

119897119905+1

= 119876119905119897119905

119878119905+1

= 119909119905+1

minus 119897119905+1

119905 = 0 1 119879 minus 1

(9)

Since Σ1




1198771015840

119879minus1)

that Σ1





119905gt 0



119906lowast

119905=

(Σ1

119905)minus1

119868

119886119905

119909119905+

1205821

119905

2(Σ1

119905)minus1

(120583119905minus 119902119905119868)

+ (Σ1

119905)minus1

(Σ0

119905minus 120574119905119868 + 120574119905+1

119864 (119876119905119877119905)) 119897119905

for 119905 = 0 119879 minus 1

(10)




119879minus1= 120582119879minus1

1205821

119905= 120582119905+ 1205821

119905+1119902119905+1

Σ1


Σ1

119905= Σ119905+ 119864 (119877

1199051198771015840

119905)119886119905+1

119886119879minus1

= 1198681015840(Σ1

119879minus1)minus1

119868 119886119905= 1198681015840(Σ1

119905)minus1

119868

119887119879minus1

= 1198681015840(Σ1

119879minus1)minus1

120583119879minus1

119887119905= 1198681015840(Σ1

119905)minus1

120583119905

119888119879minus1

= 1205831015840

119879minus1(Σ1

119879minus1)minus1

120583119879minus1

119888119905= 1205831015840

119905(Σ1

119905)minus1

120583119905

119889119879minus1

= 1198681015840(Σ1

119879minus1)minus1

Σ0

119879minus1119889119905= 1198681015840(Σ1

119905)minus1

Σ0

119905

119902119879minus1

= 119887119879minus1

119886119879minus1

119902119905= 119887119905119886119905

120572119879minus1

= 119888119879minus1


119887119879minus1

120572119905= 119888119905minus 119902119905119887119905

120574119879minus1

= 119889119879minus1

119886119879minus1

120574119905= 119889119905119886119905

120574119879minus1

= 120574119879minus1

120574119905= 120574119905+ 120574119905+1

119864(1198761199051198771015840

119905)(Σ1

119905)minus1

119868119886119905

1205731119879minus1

= 120582119879minus1

119864(119876119879minus1

) minus 120582119879minus1

(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1

1205731119905

= minus1205821

1199051205831015840

119905(Σ1

119905)minus1

(Σ0

119905minus 120574119905119868) minus 120582

1

119905120574119905+1

1205831015840

119905(Σ1

119905)minus1

119864(119876119905119877119905)

+ 120582119879minus1

119887119879minus1

120574119879minus1

+ (120582119905+ 1205731119905+1

)119864(119876119905)

1205732119879minus1

= Var(119876119879minus1

) minus (Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0

119879minus1+ 120574119879minus1

119889119879minus1

1205732119905

= [Var(119876119905) + 1205732119905+1

119864(1198762

119905)] + 119886

1199051205742

119905minus [Σ0

119905+ 120574119905+1

119864(119876119905119877119905)]1015840

sdot (Σ1

119905)minus1[Σ0

119905+ 120574119905+1

119864(119876119905119877119905)]




min Var119879minus1

(119878119879) minus 120582119879minus1

119864119879minus1

(119878119879)

st 1198771015840

119879minus1119906119879minus1

= 119909119879

1198681015840119906119879minus1

= 119909119879minus1

119897119879= 119876119879minus1

119897119879minus1

119878119879= 119909119879minus 119897119879

(11)


min 119881119879minus1

(119909119879minus1

119897119879minus1

)

= 1199061015840

119879minus1Σ1

119879minus1119906119879minus1

minus 2 [(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21199061015840

119879minus1] 119906119879minus1

+ Var (119876119879minus1

) 1198972

119879minus1+ 120582119879minus1

119864 (119876119879minus1

) 119897119879minus1

st 1198681015840119906119879minus1

= 119909119879minus1

(12)




119871 (119906119879minus1

) = 1199061015840

119879minus1Σ1

119879minus1119906119879minus1

minus 2 [(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] 119906119879minus1

+ Var (119876119879minus1

) 1198972

119879minus1+ 120582119879minus1

119864 (119876119879minus1

) 119897119879minus1

minus 120596119879(119909119879minus1

minus 1198681015840119906119879minus1

)

(13)


2Σ1

119879minus1119906119879minus1

minus 2(Σ0

119879minus1119897119879minus1

+120582119879minus1

2120583119879minus1

) + 120596119879119868 = 0 (14)

119909119879minus1

minus 1198681015840119906119879minus1

= 0 (15)


119906lowast

119879minus1= (Σ1

119879minus1)minus1

(Σ0

119879minus1119897119879minus1

+120582119879minus1

2120583119879minus1

) minus1

2120596119879(Σ1

119879minus1)minus1

119868

(16)

1198681015840119906lowast

119879minus1= 1198681015840(Σ1

119879minus1)minus1

(Σ0

119879minus1119897119879minus1

+120582119879minus1

2120583119879minus1

)

minus1

21205961198791198681015840(Σ1

119879minus1)minus1

119868

= 119889119879minus1

119897119879minus1

+120582119879minus1

2119887119879minus1

minus1

2119886119879minus1

120596119879

= 119909119879minus1

(17)


120596119879= 2

119889119879minus1

119886119879minus1

119897119879minus1

+ 120582119879minus1

119887119879minus1

119886119879minus1

minus 21

119886119879minus1

119909119879minus1

= 2120574119879minus1

119897119879minus1

+ 120582119879minus1

119902119879minus1

minus 21

119886119879minus1

119909119879minus1

(18)


119906lowast

119879minus1=

(Σ1

119879minus1)minus1

119868

119886119879minus1

119909119879minus1

+ (Σ1

119879minus1)minus1

[Σ0


119868] 119897119879minus1

+120582119879minus1

2(Σ1

119879minus1)minus1

[120583119879minus1


119868]

(19)





119881lowast

119879minus1(119909119879minus1

119897119879minus1

) =1

119886119879minus1

1199092


119902119879minus1

119909119879minus1


119909119879minus1

119897119879minus1

+ 1205731119879minus1

119897119879minus1

+ 1205732119879minus1

1198972

119879minus1minus

(120582119879minus1

)2

4120572119879minus1

(20)




min Var119879minus2

(119878119879minus1

) minus 120582119879minus2

119864119879minus2

(119878119879minus1

)

+ 119864119879minus2

(119881lowast

119879minus1(119909119879minus1

119897119879minus1

))

st 1198771015840

119879minus2119906119879minus2

= 119909119879minus1

1198681015840119906119879minus2

= 119909119879minus2

119897119879minus1

= 119876119879minus2

119897119879minus2

119878119879minus1

= 119909119879minus1


(21)


119864119879minus2

(119881lowast

119879minus1(119909119879minus1

119897119879minus1

))

=1

119886119879minus1

119864119879minus2

(1199092


119902119879minus1

119864119879minus2

(119909119879minus1

)


119864119879minus2

(119909119879minus1

119897119879minus1

) + 1205731119879minus1

119864119879minus2

(119897119879minus1

)

+ 1205732119879minus1

119864119879minus2

(1198972

119879minus1) minus

(120582119879minus1

)2

4120572119879minus1

(22)


(119881lowast

119879minus1(119909119879minus1

119897119879minus1


min 119881119879minus2

(119909119879minus2

119897119879minus2

)

st 1198681015840119906119879minus2

= 119909119879minus2

(23)

where

119881119879minus2

(119909119879minus2

119897119879minus2

) = 1199061015840

119879minus2Σ1

119879minus2119906119879minus2

minus 2 [(Σ0

119879minus2)1015840

+ 120574119879minus1

119864 (119876119879minus2

1198771015840

119879minus2)] 119897119879minus2

+1205821

119879minus2

21205831015840

119879minus2119906119879minus2

+ [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1


)2

4120572119879minus1

(24)



119871 (119906119879minus2

) = 1199061015840

119879minus2Σ1

119879minus2119906119879minus2

minus 2Φ1015840

119879minus2119906119879minus2

+ Ψ119879minus2


(119909119879minus2

minus 1198681015840119906119879minus2

)

(25)

where

Φ119879minus2

= [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)] 119897119879minus2

+1205821

119879minus2

2120583119879minus2

Ψ119879minus2

= [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1


)2

4120572119879minus1

(26)


2Σ1

119879minus2119906119879minus2


+ 120596119879minus1

119868 = 0 (27)

119909119879minus2

minus 1198681015840119906119879minus2

= 0 (28)

From (27) we have

119906lowast

119879minus2= (Σ1

119879minus2)minus1

Φ119879minus2

minus1

2120596119879minus1

(Σ1

119879minus2)minus1

119868 (29)

1198681015840119906lowast

119879minus2= 1198681015840(Σ1

119879minus2)minus1

Φ119879minus2

minus1

2120596119879minus1

1198681015840(Σ1

119879minus2)minus1

119868

= 1198681015840(Σ1

119879minus2)minus1

Φ119879minus2

minus1

2120596119879minus1

119886119879minus2

= 119909119879minus2

(30)

which implies

120596119879minus1

= 21198681015840(Σ1

119879minus2)minus1

Φ119879minus2

119886119879minus2

minus 21

119886119879minus2

119909119879minus2

(31)


into (29) we get

119906lowast

119879minus2=

(Σ1

119879minus2)minus1

119868

119886119879minus2

119909119879minus2

+ (Σ1

119879minus2)minus1

Φ119879minus2

minus1198681015840(Σ1

119879minus2)minus1

Φ119879minus2

119886119879minus2

(Σ1

119879minus2)minus1

119868

(32)



119906lowast

119879minus2=

(Σ1

119879minus2)minus1

119868

119886119879minus2

119909119879minus2

+1205821

119879minus2

2(Σ1

119879minus2)minus1

(120583119879minus2


119868)

+ (Σ1

119879minus2)minus1

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868] 119897119879minus2

(33)




details)

119881lowast

119879minus2(119909119879minus2

119897119879minus2

) =1

119886119879minus2

1199092


119879minus2119902119879minus2

119909119879minus2


119909119879minus2

119897119879minus2

+ 1205731119879minus2

119897119879minus2

+ 1205732119879minus2

1198972

119879minus2


)2

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(34)


119881lowast

119905(119909119905 119897119905) =

1

119886119905

1199092

119905minus 1205821

119905119902119905119909119905minus 2120574119905119909119905119897119905

+ 1205731119905119897119905+ 12057321199051198972

119905minus

119879minus1

sum119894=119905

(1205821

119894)2

4120572119894

(35)


119905minus1and



min Var119905minus1

(119878119905) minus 120582119905minus1

119864119905minus1

(119878119905)

+ 119864119905minus1

(119881lowast

119905(119909119905 119897119905))

st 1198771015840

119905minus1119906119905minus1

= 119909119905

1198681015840119906119905minus1

= 119909119905minus1

119897119905= 119876119905minus1

119897119905minus1

119878119905= 119909119905minus 119897119905

(36)


119864119905minus1

(119881lowast

119905(119909119905 119897119905)) =

1

119886119905

119864119905minus1

(1199092

119905) minus 1205821

119905119902119905119864119905minus1

(119909119905)

minus 2120574119905119864119905minus1

(119909119905119897119905) + 1205731119905119864119905minus1

(119897119905)

+ 1205732119905119864119905minus1

(1198972

119905) minus

119879minus1

sum119894=119905

(1205821

119894)2

4120572119894

(37)


(119881lowast



min 119881119905minus1

(119909119905minus1

119897119905minus1

) = 1199061015840

119879minus1Σ1

119905minus1119906119905minus1

minus 2Φ1015840

119905minus1119906119905minus1

+ Ψ119905minus1

st 1198681015840119906119905minus1

= 119909119905minus1

(38)

where

Φ119905minus1

= [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] 119897119905minus1

+1205821

119905minus1

2120583119905minus1

Ψ119905minus1

= [Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] 1198972

119905minus1

+ [120582119905minus1

+ 1205731119905] 119864 (119876

119905minus1) 119897119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4120572119894

(39)



119871 (119906119905minus1

) = 1199061015840

119879minus1Σ1

119905minus1119906119905minus1

minus 2Φ1015840

119905minus1119906119905minus1

+ Ψ119905minus1

minus 120596119905(119909119905minus1

minus 1198681015840119906119905minus1

)

(40)


2Σ1

119905minus1119906119905minus1


+ 120596119905119868 = 0

119909119905minus1

minus 1198681015840119906119905minus1

= 0

(41)

Thus we have

119906lowast

119905minus1= (Σ1

119905minus1)minus1

Φ119905minus1

minus1

2120596119905(Σ1

119905minus1)minus1

119868 (42)

1198681015840119906lowast

119905minus1= 1198681015840(Σ1

119905minus1)minus1

Φ119905minus1

minus1

21205961199051198681015840(Σ1

119905minus1)minus1

119868

= 1198681015840(Σ1

119905minus1)minus1

Φ119905minus1

minus1

2120596119905119886119905minus1

= 119909119905minus1

(43)

which implies

120596119905= 2

1198681015840(Σ1

119905minus1)minus1

Φ119905minus1

119886119905minus1

minus 21

119886119905minus1

119909119905minus1

(44)


119906lowast

119905minus1=

(Σ1

119905minus1)minus1

119868

119886119905minus1

119909119905minus1

minus1198681015840(Σ1

119905minus1)minus1

Φ119905minus1

119886119905minus1

(Σ1

119905minus1)minus1

119868

+ (Σ1

119905minus1)minus1

Φ119905minus1

=(Σ1

119905minus1)minus1

119868

119886119905minus1

119909119905minus1

+1205821

119905minus1

2(Σ1

119905minus1)minus1

(120583119905minus1


119868)

+ (Σ1

119905minus1)minus1

[Σ0


119868 + 120574119905119864 (119876119905minus1

119877119905minus1

)] 119897119905minus1

(45)




119881lowast

119905minus1(119909119905minus1

119897119905minus1

) =1

119886119905minus1

1199092


119905minus1119902119905minus1

119909119905minus1


119909119905minus1

119897119905minus1

+ 1205731119905minus1

119897119905minus1

+ 1205732119905minus1

1198972

119905minus1minus

119879minus1

sum119894=119905minus1

(1205821

119894)2

4120572119894





119906lowast

119905=

(Σ1

119905)minus1

119868

119886119905

119909119905+

1205821

119905

2(Σ1

119905)minus1

(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1

(47)





119906lowast

119905=

(Σ1

119905)minus1

119868

119886119905

119909119905+

1205821

119905

2(Σ1

119905)minus1

(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1

(48)


119905= 1198681015840(Σ1

119905)minus1Σ0

119905=

0 120574119905= 119889119905119886119905= 0 and 120574


119906lowast

119905=

(Σ1

119905)minus1

119868

119886119905

119909119905+

1205821

119905

2(Σ1

119905)minus1

(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1

(49)




(Σ1

119905)minus1

(Σ0

119905minus 120574119905119868 + 120574119905+1

119864 (119876119905119877119905)) 119897119905 (50)



119905


min119906119905

Var119905(119878119905+1

) minus 120582119905119864119905(119878119905+1

)

st 1198771015840

119905119906119905= 119909119905+1

1198681015840119906119905= 119909119905

119897119905+1

= 119876119905119897119905

119878119905+1

= 119909119905+1

minus 119897119905+1

(51)


119906119898119910

119905=

(Σ119905)minus1

119868

119886119905

119909119905+

120582119905

2(Σ119905)minus1

(120583119905minus 119902119905119868)

+ (Σ119905)minus1

(Σ0

119905minus 120574119905119868) 119897119905 119905 = 0 119879 minus 1

(52)


119905119886119905


119905rarr 0


119906lowast

119905=

(Σ1

119905)minus1

119868

119886119905

119909119905+ (Σ1

119905)minus1

(Σ0

119905minus 120574119905119868 + 120574119905+1

119864 (119876119905119877119905)) 119897119905

119905 = 0 119879 minus 1

119906119898119910

119905=

(Σ119905)minus1

119868

119886119905

119909119905+ (Σ119905)minus1

(Σ0

119905minus 120574119905119868) 119897119905 119905 = 0 119879 minus 1

(53)







119909119905= 1198751015840

119905minus1119906119905minus1

+ 119904119905minus1

119909119905minus1

(54)



0(1198781) + 1198640(1205881(1198782) + 1198641(1205882(1198783) + sdot sdot sdot

+119864119879minus3

(120588119879minus2

(119878119879minus1

) + 119864119879minus2

(120588119879minus1

(119878119879))) ))

st 119909119905+1

= 1198751015840

119905119906119905+ 119904119905119909119905

119897119905+1

= 119876119905119897119905

119878119905+1

= 119909119905+1

minus 119897119905+1

119905 = 0 1 119879 minus 1

(55)



119906lowast

119905= Σminus1

119905120593119905119897119905+

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (56)

where 120593119905= 119864(119875

119905119876119905) minus 119864(119875

119905)119864(119876119905) and 120598

119905= 119864(119875

119905)




119879minus1= 120582119879minus1

1205821

119905= 120582119905+ 1205821

119905+1119904119905+1

1205731119879minus1

= 120582119879minus1

[119864 (119876119879minus1

) minus 1205981015840


)minus1

120593119879minus1

] 1205731119905

= (120582119905+ 1205731119905+1

) 119864(119876119905) minus 1205821

1199051205981015840

119905(Σ119905)minus1120593119905

1205732119879minus1

= Var(119876119879minus1

) minus 1205931015840


)minus1

120593119879minus1

1205732119905

= Var(119876119905) + 1205732119905+1

119864(1198762

119905) minus 1205931015840

119905(Σ119905)minus1120593119905




min119906119879minus1

Var119879minus1

(119878119879) minus 120582119879minus1

119864119879minus1

(119878119879)

st 119909119879

= 119904119879minus1

119909119879minus1

+ 1198751015840

119879minus1119906119879minus1

119897119879

= 119876119879minus1

119897119879minus1

119878119879= 119909119879minus 119897119879

(57)


min119906119879minus1

119881119879minus1

(119909119879minus1

119897119879minus1

) (58)

where

119881119879minus1

(119909119879minus1

119897119879minus1

) = 1199061015840


119906119879minus1

minus 2(1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840

119879minus1)119906119879minus1

+ Var (119876119879minus1

) 1198972


119904119879minus1

119909119879minus1

+ 120582119879minus1

119864 (119876119879minus1

) 119897119879minus1

(59)


119889 (119881119879minus1

(119909119879minus1

119897119879minus1

))

119889 (119906119879minus1

)= 2Σ119879minus1

119906119879minus1

minus 2(120593119879minus1

119897119879minus1

+120582119879minus1

2120598119879minus1

)

= 0

(60)

which implies

119906lowast

119879minus1= (Σ119879minus1

)minus1

[120593119879minus1

119897119879minus1

+120582119879minus1

2120598119879minus1

] (61)




119881lowast

119879minus1(119909119879minus1

119897119879minus1

) = 1205732119879minus1

1198972

119879minus1+ 1205731119879minus1

119897119879minus1


119904119879minus1

119909119879minus1


)2

4119872119879minus1

(62)


= 1205981015840


119879minus1120598119879minus1




min Var119879minus2

(119878119879minus1

) minus 120582119879minus2

119864119879minus2

(119878119879minus1

)

+ 119864119879minus2

(119881lowast

119879minus1(119909119879minus1

119897119879minus1

))

st 119909119879minus1

= 119904119879minus2

119909119879minus2

+ 1198751015840

119879minus2119906119879minus2

119897119879minus1

= 119876119879minus2

119897119879minus2

119878119879minus1

= 119909119879minus1


(63)


119864119879minus2

(119881lowast

119879minus1(119909119879minus1

119897119879minus1

)) = 1205732119879minus1

119864 (1198762

119879minus2) 1198972

119879minus2

+ 1205731119879minus1

119864 (119876119879minus2

) 119897119879minus2


119904119879minus1

119904119879minus2

119909119879minus2


119904119879minus1

1205981015840

119879minus2119906119879minus2


)2

4119872119879minus1

(64)


(119881lowast

119879minus1(119909119879minus1

119897119879minus1


min119906119879minus2

119881119879minus2

(119909119879minus2

119897119879minus2

) (65)


(119909119879minus2

119897119879minus2

) = 1199061015840


119906119879minus2

minus 2(1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840

119879minus2)119906119879minus2

+ [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

(66)


119889 (119881119879minus2

(119909119879minus2

119897119879minus2

))

119889 (119906119879minus2

)= 2Σ119879minus2

119906119879minus2

minus 2(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= 0

(67)


Thus

119906lowast

119879minus2= (Σ119879minus2

)minus1

(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= (Σ119879minus2

)minus1

120593119879minus2

119897119879minus2

+1205821

119879minus2

2(Σ119879minus2

)minus1

120598119879minus2

(68)




119881lowast

119879minus2(119909119879minus2

119897119879minus2

) = 1205732119879minus2

1198972

119879minus2+ 1205731119879minus2

119897119879minus2

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

minus(1205821

119879minus2)2

4119872119879minus2

(69)


= 1205981015840


119879minus2120598119879minus2


both (56) and

119881lowast

119905(119909119905 119897119905) = 12057321199051198972

119905+ 1205731119905119897119905minus 1205821

119905119904119905119909119905minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

(70)

hold where 119872119894= 1205981015840

119894Σminus1



119905minus1and



min Var119905minus1

(119878119905) minus 120582119905minus1

119864119905minus1

(119878119905) + 119864119905minus1

(119881lowast

119905(119909119905 119897119905))

st 119909119905= 119904119905minus1

119909119905minus1

+ 1198751015840

119905minus1119906119905minus1

119897119905= 119876119905minus1

119897119905minus1

119878119905= 119909119905minus 119897119905

(71)


119864119905minus1

(119881lowast

119905(119909119905 119897119905)) = 120573

2119905119864 (1198762

119905minus1) 1198972

119905minus1+ 1205731119905119864 (119876119905minus1

) 119897119905

minus 1205821

119905119904119905119904119905minus1

119909119905minus1

minus 1205821

1199051199041199051205981015840

119905minus1119906119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

(72)


(119881lowast



119881119905minus1

(119909119905minus1

119897119905minus1

) (73)

where

119881119905minus1

(119909119905minus1

119897119905minus1

) = 1199061015840


119906119905minus1

minus 2(1205931015840

119905minus1119897119905minus1

+1205821

119905minus1

21205981015840

119905minus1)119906119905minus1

+ [Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] 1198972

119905minus1

+ [120582119905minus1

119864 (119876119905minus1

) + 1205731119905119864 (119876119905minus1

)] 119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

(74)


119889 (119881119905minus1

(119909119905minus1

119897119905minus1

))

119889 (119906119905minus1

)= 2Σ119905minus1

119906119905minus1

minus 2(120593119905minus1

119897119905minus1

+1205821

119905minus1

2120598119905minus1

)

= 0

(75)

which implies

119906lowast


119905minus1120593119905minus1

119897119905minus1

+1205821

119905minus1

2Σminus1

119905minus1120598119905minus1

(76)




119881lowast

119905minus1(119909119905minus1

119897119905minus1

) = 1205732119905minus1

1198972

119905minus1+ 1205731119905minus1

119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905minus1

(1205821

119894)2

4119872119894

(77)







119906lowast

119905=

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (78)






119906lowast

119905=

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (79)


120593119905= 119864 (119875

119905119876119905) minus 119864 (119875

119905) 119864 (119876

119905)

= 119864 ((119877119905minus 119904119905119868)119876119905) minus 119864 (119877

119905minus 119904119905119868) 119864 (119876

119905)

= 119864 (119877119905119876119905) minus 119864 (119877

119905) 119864 (119876

119905)

= Σ0

119905

= 0

(80)



119906lowast

119905=

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (81)




Σminus1

119905120593119905119897119905 (82)



min119906119905

Var119905(119878119905+1

) minus 120582119905119864119905(119878119905+1

)

st 119909119905+1

= 119904119905119909119905+ 1198751015840

119905119906119905

119897119905+1

= 119876119905119897119905

119878119905+1

= 119909119905+1

minus 119897119905+1

(83)


119906119898119910

119905= Σminus1

119905120593119905119897119905+

120582119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (84)




119906lowast

119905= Σminus1

119905120593119905119897119905 119905 = 0 119879 minus 1 (85)






119905


lowast

119905

0 (

11941198970+ 0903119909

0+ 3736

01071198970minus 0066119909

0+ 0203

minus05771198970+ 0163119909

0minus 3939

) (

09031199090+ 3736

minus00661199090+ 0203

01631199090minus 3939

)

1 (

minus03721198971+ 1018119909

1+ 1743

01981198971minus 0060119909

1+ 0095

minus06331198971+ 0042119909

1minus 1838

) (

10181199091+ 1743

minus00601199091+ 0095

00421199091minus 1838

)




my119905

0 (

09031199090+ 1194119897

0+ 3736

minus00661199090+ 0107119897

0+ 0203

01631199090minus 0577119897

0minus 3939

) (

10181199090+ 0449119897

0+ 1743

minus00601199090+ 0150119897

0+ 0095

00421199090minus 0599119897

0minus 1838

)

1 (

10181199091minus 0372119897

1+ 1743

minus00601199091+ 0198119897

1+ 0095

00421199091minus 0633119897

1minus 1838

) (

10181199091+ 0449119897

1+ 1743

minus00601199091+ 0150119897

1+ 0095

00421199091minus 0599119897

1minus 1838

)



119905=

(1162 1246 1228) and

Σ119905= (

00146 00187 00145

00187 00854 00104

00145 00104 00289

) (86)






119905=

05 and 120582119905







my119905

0 (

09031199090+ 1194119897

0

minus00661199090+ 0107119897

0

01631199090minus 0577119897

0

) (

10181199090+ 0449119897

0

minus00601199090+ 0150119897

0

00421199090minus 0599119897

0

)

1 (

10181199091minus 0372119897

1

minus00601199091+ 0198119897

1

00421199091minus 0633119897

1

) (

10181199091+ 0449119897

1

minus00601199091+ 0150119897

1

00421199091minus 0599119897

1

)




my119905

0 (

04441198970+ 7889

01501198970minus 0062

minus05991198970minus 3572

) (

04441198970+ 3867

01501198970minus 0031

minus05991198970minus 1751

)

1 (

04441198971+ 3867

01501198971minus 0031

minus05991198971minus 1751

) (

04441198971+ 3867

01501198971minus 0031

minus05991198971minus 1751

)




my119905

0 (

04441198970

01501198970

minus05991198970

) (

04441198970

01501198970

minus05991198970

)

1 (

04441198971

01501198971

minus05991198971

) (

04441198971

01501198971

minus05991198971

)


119905= 120583119905minus 119904119905119868 =

(0122 0206 0188)1015840 and120593

119905= Σ0

119905= (00006 00149 00050)





6 Conclusion


Appendices

A Proof of (20)


119879minus1into 119906

lowast

119879minus1

1015840Σ1


119879minus1gives

119906lowast

119879minus1

1015840

Σ1


119879minus1=

119868

119886119879minus1

119909119879minus1

+ [Σ0


119868] 119897119879minus1

+120582119879minus1

2[120583119879minus1


119868]

1015840

(Σ1

119879minus1)minus1

times 119868

119886119879minus1

119909119879minus1

+ [Σ0


119868]


+120582119879minus1

2[120583119879minus1


119868]

=1

119886119879minus1

1199092

119879minus1+ [Σ0


119868]1015840

times (Σ1

119879minus1)minus1

[Σ0


119868] 1198972

119879minus1

+ 120582119879minus1

[Σ0


119868]1015840

times (Σ1

119879minus1)minus1

[120583119879minus1


119868] 119897119879minus1

+(120582119879minus1

)2

4[120583119879minus1


119868]1015840

times (Σ1

119879minus1)minus1

[120583119879minus1


119868]

=1

119886119879minus1

1199092

119879minus1+ [(Σ

0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0

119879minus1


1205742

119879minus1] 1198972

119879minus1

+ [(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1


120574119879minus1

]


119897119879minus1

+(120582119879minus1

)2

4120572119879minus1

(A1)



0

119879minus1)1015840

119897119879minus1

+

(120582119879minus1

2)1205831015840



2 [(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840


119879minus1

=2

119886119879minus1

[(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1

119868119909119879minus1

+ 2 [(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1


times [Σ0


119868] 119897119879minus1

+ 120582119879minus1

[(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1



119868]

= 2120574119879minus1

119897119879minus1

119909119879minus1

+ 119902119879minus1

120582119879minus1

119909119879minus1

+ 2 [(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0


119889119879minus1

] 1198972

119879minus1

+ 2120582119879minus1

[(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1


120574119879minus1

] 119897119879minus1

+(120582119879minus1

)2

2120572119879minus1

(A2)

Thus119881lowast

119879minus1(119909119879minus1

119897119879minus1

)

=1

119886119879minus1

1199092


119902119879minus1

119909119879minus1


119909119879minus1

119897119879minus1

+ 120582119879minus1

[119864 (119876119879minus1

) minus (Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1

+119887119879minus1

120574119879minus1

] 119897119879minus1

+ [Var (119876119879minus1

) minus (Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0

119879minus1

+120574119879minus1

119889119879minus1

] 1198972

119879minus1minus

(120582119879minus1

)2

4120572119879minus1

=1

119886119879minus1

1199092


119902119879minus1

119909119879minus1


119909119879minus1

119897119879minus1

+ 1205731119879minus1

119897119879minus1

+ 1205732119879minus1

1198972

119879minus1minus

(120582119879minus1

)2

4120572119879minus1

(A3)


1198681015840

119886119879minus2

(Σ1

119879minus2)minus1

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

=119889119879minus2

119886119879minus2

+ 120574119879minus1

1198681015840(Σ1

119879minus2)minus1

119864 (119876119879minus2

119877119879minus2

)

119886119879minus2


= 120574119879minus2

+ 120574119879minus1

1198681015840(Σ1

119879minus2)minus1

119864 (119876119879minus2

119877119879minus2

)

119886119879minus2


= 120574119879minus2


= 0

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1

[120583119879minus2


119868] =119887119879minus2

119886119879minus2


= 0

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

119868

= 119886119879minus2

120574119879minus2

+ 120574119879minus1

119864 (119876119879minus2

1198771015840

119879minus2) (Σ1

119879minus2)minus1

119868

119886119879minus2

= 119886119879minus2

120574119879minus2

(B1)


119879minus2

1015840Σ1




and 119897119879minus2

119909119879minus2

in 119906lowast

119879minus2

1015840Σ1



119879minus2is

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1 119868

119886119879minus2

=1

119886119879minus2

(B2)


is

1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

= 1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

minus 1205821

119879minus2120574119879minus2

1198681015840(Σ1

119879minus2)minus1

[120583119879minus2


119868]

= 1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

(B3)


is

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]1015840

sdot (Σ1

119879minus2)minus1

sdot [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

= [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]


1205742

119879minus2

(B4)


(1205821

119879minus2)2

4[120583119879minus2


119868]1015840

(Σ1

119879minus2)minus1

[120583119879minus2


119868]

=(1205821

119879minus2)2

4120572119879minus2

(B5)




119879minus2


is

1205821

119879minus2

2

1205831015840

119879minus2(Σ1

119879minus2)minus1

119868

119886119879minus2

=1205821

119879minus2119902119879minus2

2 (B6)



119909119879minus2

is 120574119879minus2


is

1205821

119879minus2

2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

(120583119879minus2


119868)

+1205821

119879minus2

21205831015840

119879minus2(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

(B7)


is

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

= [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)] minus 119886119879minus2

1205742

119879minus2

(B8)


(1205821

119879minus2

2)

2

1205831015840

119879minus2(Σ1

119879minus2)minus1

(120583119879minus2


119868) = (1205821

119879minus2

2)

2

120572119879minus2

(B9)


119879minus2(119909119879minus2

119897119879minus2


119879minus2is

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1 119868

119886119879minus2

=1

119886119879minus2

(B10)



119902119879minus2


119909119879minus2



is

[Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] + 119886

119879minus21205742

119879minus2

minus [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]

= 1205732119879minus2

(B11)


is

minus 1205821

119879minus21205831015840

119879minus2(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)]

= 1205731119879minus2

(B12)


minus1205822

119879minus1

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(B13)

Thus

119881lowast

119879minus2(119909119879minus2

119897119879minus2

) =1

119886119879minus2

1199092


119879minus2119902119879minus2

119909119879minus2


119909119879minus2

119897119879minus2

+ 1205731119879minus2

119897119879minus2

+ 1205732119879minus2

1198972

119879minus2minus

1205822

119879minus1

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(B14)


1198681015840

119886119905minus1

(Σ1

119905minus1)minus1

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

=119889119905minus1

119886119905minus1

+ 120574119905

1198681015840(Σ1

119905minus1)minus1

119864 (119876119905minus1

119877119905minus1

)

119886119905minus1


= 120574119905minus1

+ 120574119905

1198681015840(Σ1

119905minus1)minus1

119864 (119876119905minus1

119877119905minus1

)

119886119905minus1


= 120574119905minus1


= 0

1198681015840

119886119905minus1

(Σ1

119905minus1)minus1

[120583119905minus1


119868] =119887119905minus1

119886119905minus1


= 0

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

119868

= 119886119905minus1

120574119905minus1

+ 120574119905

119864 (119876119905minus1

1198771015840

119905minus1) (Σ1

119905minus1)minus1

119868

119886119905minus1

= 119886119905minus1

120574119905minus1

(C1)


119905minus1

1015840Σ1




and 119897119905minus1

119909119905minus1


is

1198681015840

119886119905minus1

(Σ1

119905minus1)minus1 119868

119886119905minus1

=1

119886119905minus1

(C2)


is

1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

= 1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

minus 1205821

119905minus1120574119905minus1

1198681015840(Σ1

119905minus1)minus1

[120583119905minus1


119868]


= 1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

(C3)


is

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]1015840

sdot (Σ1

119905minus1)minus1

sdot [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

= [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] minus 119886119905minus1

1205742

119905minus1

(C4)


(1205821

119905minus1)2

4[120583119905minus1


119868]1015840

(Σ1

119905minus1)minus1

[120583119905minus1


119868] =(1205821

119905minus1)2

4120572119905minus1

(C5)




119905minus1is


is

1205821

119905minus1

2

1205831015840

119905minus1(Σ1

119905minus1)minus1

119868

119886119905minus1

=1205821

119905minus1119902119905minus1

2 (C6)


119909119905minus1

is 120574119905minus1


is

1205821

119905minus1

2[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1



119868) +1205821

119905minus1

21205831015840

119905minus1(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

(C7)


is

[Σ0

119905minus1+ 120574119905119864(119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

= [Σ0

119905minus1+ 120574119905119864(119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] minus 119886119905minus1

1205742

119905minus1

(C8)


(1205821

119905minus1

2)

2

120583119905minus1

(Σ1

119905minus1)minus1

(120583119905minus1


119868) = (1205821

119905minus1

2)

2

120572119905minus1

(C9)


119905minus1(119909119905minus1

119897119905minus1


119905minus1is 1119886



1

119905minus1119902119905minus1


119905minus1119909119905minus1



is

[Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] + 119886

119905minus11205742

119905minus1

minus [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]

= 1205732119905minus1

(C10)


is

minus 1205821

119905minus11205831015840

119905minus1(Σ1

119905minus1)minus1

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

+ (120582119905minus1

+ 1205731119905) 119864 (119876

119905minus1) = 1205731119905minus1

(C11)


(1205821

119894)2120572119894

Thus

119881lowast

119905minus1(119909119905minus1

119897119905minus1

) =1

119886119905minus1

1199092


119905minus1119902119905minus1

119909119905minus1


119909119905minus1

119897119905minus1

+ 1205731119905minus1

119897119905minus1

+ 1205732119905minus1

1198972

119905minus1minus

1

4

119879minus1

sum119894=119905minus1

(1205821

119894)2

120572119894

(C12)

D Proof of (62)


119879minus1into 119906

lowast

119879minus1

1015840Σ119879minus1

119906lowast


119906lowast

119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1= (1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840

119879minus1)


)minus1

(120593119879minus1

119897119879minus1

+120582119879minus1

2120598119879minus1

)

= 1205931015840


)minus1

120593119879minus1

1198972

119879minus1+ 120582119879minus1

1205981015840

119879minus1


)minus1

120593119879minus1

119897119879minus1

+(120582119879minus1

)2

4119872119879minus1

2 (1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840



119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1

(D1)

Thus

119881lowast

119879minus1(119909119879minus1

119897119879minus1


119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1+ Var (119876

119879minus1) 1198972

119879minus1


119904119879minus1

119909119879minus1

+ 120582119879minus1

119864 (119876119879minus1

) 119897119879minus1

= [Var (119876119879minus1

) minus 1205931015840


)minus1

120593119879minus1

] 1198972

119879minus1

+ 120582119879minus1

[119864 (119876119879minus1

) minus 1205981015840


)minus1

120593119879minus1

]



119904119879minus1

119909119879minus1


)2

4119872119879minus1

= 1205732119879minus1

1198972

119879minus1+ 1205731119879minus1

119897119879minus1


119904119879minus1

119909119879minus1


)2

4119872119879minus1

(D2)



lowast


119906lowast

119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2= (1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840

119879minus2)


)minus1

(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= 1205931015840


)minus1

120593119879minus2

1198972

119879minus2+ 1205821

119879minus21205981015840

119879minus2


)minus1

120593119879minus2

119897119879minus2

+(1205821

119879minus2)2

4119872119879minus2

2 1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840



119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2

(E1)Thus

119881lowast

119879minus2(119909119879minus2

119897119879minus2


119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2

+ [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

= 1205732119879minus2

1198972

119879minus2+ 1205731119879minus2

119897119879minus2

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

minus(1205821

119879minus2)2

4119872119879minus2

(E2)where1205731119879minus2

= (120582119879minus2

+ 1205731119879minus1

) 119864 (119876119879minus2

) minus 1205821

119879minus21205981015840


)minus1

120593119879minus2

1205732119879minus2

= Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2) minus 1205931015840


)minus1

120593119879minus2

(E3)


lowast

119905minus1that

119906lowast

119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1= (1205931015840

119879minus1119897119905minus1

+1205821

119905minus1

21205981015840

119879minus1)


)minus1

(120593119905minus1

119897119905minus1

+1205821

119905minus1

2120598119905minus1

)

= 1205931015840


)minus1

120593119905minus1

1198972

119905minus1+ 1205821

119905minus11205981015840

119879minus1


)minus1

120593119905minus1

119897119905minus1

+(1205821

119905minus1)2

4119872119905minus1

2 (1205931015840

119879minus1119897119905minus1

+1205821

119905minus1

21205981015840



119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1

(F1)

Thus

119881lowast

119905minus1(119909119905minus1

119897119905minus1


119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1

+ [Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] 1198972

119905minus1

+ [120582119905minus1

119864 (119876119905minus1

) + 1205731119905119864 (119876119905minus1

)] 119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

= 1205732119905minus1

1198972

119905minus1+ 1205731119905minus1

119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905minus1

(1205821

119894)2

4119872119894

(F2)

where

1205731119905minus1

= (120582119905minus1

+ 1205731119905) 119864 (119876

119905minus1) minus 1205821

119905minus11205981015840


)minus1

120593119905minus1

1205732119905minus1

= Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1) minus 1205931015840


)minus1

120593119905minus1

(F3)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119879minus1= 120582119879minus1

1205821

119905= 120582119905+ 1205821

119905+1119902119905+1

Σ1


Σ1

119905= Σ119905+ 119864 (119877

1199051198771015840

119905)119886119905+1

119886119879minus1

= 1198681015840(Σ1

119879minus1)minus1

119868 119886119905= 1198681015840(Σ1

119905)minus1

119868

119887119879minus1

= 1198681015840(Σ1

119879minus1)minus1

120583119879minus1

119887119905= 1198681015840(Σ1

119905)minus1

120583119905

119888119879minus1

= 1205831015840

119879minus1(Σ1

119879minus1)minus1

120583119879minus1

119888119905= 1205831015840

119905(Σ1

119905)minus1

120583119905

119889119879minus1

= 1198681015840(Σ1

119879minus1)minus1

Σ0

119879minus1119889119905= 1198681015840(Σ1

119905)minus1

Σ0

119905

119902119879minus1

= 119887119879minus1

119886119879minus1

119902119905= 119887119905119886119905

120572119879minus1

= 119888119879minus1


119887119879minus1

120572119905= 119888119905minus 119902119905119887119905

120574119879minus1

= 119889119879minus1

119886119879minus1

120574119905= 119889119905119886119905

120574119879minus1

= 120574119879minus1

120574119905= 120574119905+ 120574119905+1

119864(1198761199051198771015840

119905)(Σ1

119905)minus1

119868119886119905

1205731119879minus1

= 120582119879minus1

119864(119876119879minus1

) minus 120582119879minus1

(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1

1205731119905

= minus1205821

1199051205831015840

119905(Σ1

119905)minus1

(Σ0

119905minus 120574119905119868) minus 120582

1

119905120574119905+1

1205831015840

119905(Σ1

119905)minus1

119864(119876119905119877119905)

+ 120582119879minus1

119887119879minus1

120574119879minus1

+ (120582119905+ 1205731119905+1

)119864(119876119905)

1205732119879minus1

= Var(119876119879minus1

) minus (Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0

119879minus1+ 120574119879minus1

119889119879minus1

1205732119905

= [Var(119876119905) + 1205732119905+1

119864(1198762

119905)] + 119886

1199051205742

119905minus [Σ0

119905+ 120574119905+1

119864(119876119905119877119905)]1015840

sdot (Σ1

119905)minus1[Σ0

119905+ 120574119905+1

119864(119876119905119877119905)]




min Var119879minus1

(119878119879) minus 120582119879minus1

119864119879minus1

(119878119879)

st 1198771015840

119879minus1119906119879minus1

= 119909119879

1198681015840119906119879minus1

= 119909119879minus1

119897119879= 119876119879minus1

119897119879minus1

119878119879= 119909119879minus 119897119879

(11)


min 119881119879minus1

(119909119879minus1

119897119879minus1

)

= 1199061015840

119879minus1Σ1

119879minus1119906119879minus1

minus 2 [(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21199061015840

119879minus1] 119906119879minus1

+ Var (119876119879minus1

) 1198972

119879minus1+ 120582119879minus1

119864 (119876119879minus1

) 119897119879minus1

st 1198681015840119906119879minus1

= 119909119879minus1

(12)




119871 (119906119879minus1

) = 1199061015840

119879minus1Σ1

119879minus1119906119879minus1

minus 2 [(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] 119906119879minus1

+ Var (119876119879minus1

) 1198972

119879minus1+ 120582119879minus1

119864 (119876119879minus1

) 119897119879minus1

minus 120596119879(119909119879minus1

minus 1198681015840119906119879minus1

)

(13)


2Σ1

119879minus1119906119879minus1

minus 2(Σ0

119879minus1119897119879minus1

+120582119879minus1

2120583119879minus1

) + 120596119879119868 = 0 (14)

119909119879minus1

minus 1198681015840119906119879minus1

= 0 (15)


119906lowast

119879minus1= (Σ1

119879minus1)minus1

(Σ0

119879minus1119897119879minus1

+120582119879minus1

2120583119879minus1

) minus1

2120596119879(Σ1

119879minus1)minus1

119868

(16)

1198681015840119906lowast

119879minus1= 1198681015840(Σ1

119879minus1)minus1

(Σ0

119879minus1119897119879minus1

+120582119879minus1

2120583119879minus1

)

minus1

21205961198791198681015840(Σ1

119879minus1)minus1

119868

= 119889119879minus1

119897119879minus1

+120582119879minus1

2119887119879minus1

minus1

2119886119879minus1

120596119879

= 119909119879minus1

(17)


120596119879= 2

119889119879minus1

119886119879minus1

119897119879minus1

+ 120582119879minus1

119887119879minus1

119886119879minus1

minus 21

119886119879minus1

119909119879minus1

= 2120574119879minus1

119897119879minus1

+ 120582119879minus1

119902119879minus1

minus 21

119886119879minus1

119909119879minus1

(18)


119906lowast

119879minus1=

(Σ1

119879minus1)minus1

119868

119886119879minus1

119909119879minus1

+ (Σ1

119879minus1)minus1

[Σ0


119868] 119897119879minus1

+120582119879minus1

2(Σ1

119879minus1)minus1

[120583119879minus1


119868]

(19)





119881lowast

119879minus1(119909119879minus1

119897119879minus1

) =1

119886119879minus1

1199092


119902119879minus1

119909119879minus1


119909119879minus1

119897119879minus1

+ 1205731119879minus1

119897119879minus1

+ 1205732119879minus1

1198972

119879minus1minus

(120582119879minus1

)2

4120572119879minus1

(20)




min Var119879minus2

(119878119879minus1

) minus 120582119879minus2

119864119879minus2

(119878119879minus1

)

+ 119864119879minus2

(119881lowast

119879minus1(119909119879minus1

119897119879minus1

))

st 1198771015840

119879minus2119906119879minus2

= 119909119879minus1

1198681015840119906119879minus2

= 119909119879minus2

119897119879minus1

= 119876119879minus2

119897119879minus2

119878119879minus1

= 119909119879minus1


(21)


119864119879minus2

(119881lowast

119879minus1(119909119879minus1

119897119879minus1

))

=1

119886119879minus1

119864119879minus2

(1199092


119902119879minus1

119864119879minus2

(119909119879minus1

)


119864119879minus2

(119909119879minus1

119897119879minus1

) + 1205731119879minus1

119864119879minus2

(119897119879minus1

)

+ 1205732119879minus1

119864119879minus2

(1198972

119879minus1) minus

(120582119879minus1

)2

4120572119879minus1

(22)


(119881lowast

119879minus1(119909119879minus1

119897119879minus1


min 119881119879minus2

(119909119879minus2

119897119879minus2

)

st 1198681015840119906119879minus2

= 119909119879minus2

(23)

where

119881119879minus2

(119909119879minus2

119897119879minus2

) = 1199061015840

119879minus2Σ1

119879minus2119906119879minus2

minus 2 [(Σ0

119879minus2)1015840

+ 120574119879minus1

119864 (119876119879minus2

1198771015840

119879minus2)] 119897119879minus2

+1205821

119879minus2

21205831015840

119879minus2119906119879minus2

+ [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1


)2

4120572119879minus1

(24)



119871 (119906119879minus2

) = 1199061015840

119879minus2Σ1

119879minus2119906119879minus2

minus 2Φ1015840

119879minus2119906119879minus2

+ Ψ119879minus2


(119909119879minus2

minus 1198681015840119906119879minus2

)

(25)

where

Φ119879minus2

= [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)] 119897119879minus2

+1205821

119879minus2

2120583119879minus2

Ψ119879minus2

= [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1


)2

4120572119879minus1

(26)


2Σ1

119879minus2119906119879minus2


+ 120596119879minus1

119868 = 0 (27)

119909119879minus2

minus 1198681015840119906119879minus2

= 0 (28)

From (27) we have

119906lowast

119879minus2= (Σ1

119879minus2)minus1

Φ119879minus2

minus1

2120596119879minus1

(Σ1

119879minus2)minus1

119868 (29)

1198681015840119906lowast

119879minus2= 1198681015840(Σ1

119879minus2)minus1

Φ119879minus2

minus1

2120596119879minus1

1198681015840(Σ1

119879minus2)minus1

119868

= 1198681015840(Σ1

119879minus2)minus1

Φ119879minus2

minus1

2120596119879minus1

119886119879minus2

= 119909119879minus2

(30)

which implies

120596119879minus1

= 21198681015840(Σ1

119879minus2)minus1

Φ119879minus2

119886119879minus2

minus 21

119886119879minus2

119909119879minus2

(31)


into (29) we get

119906lowast

119879minus2=

(Σ1

119879minus2)minus1

119868

119886119879minus2

119909119879minus2

+ (Σ1

119879minus2)minus1

Φ119879minus2

minus1198681015840(Σ1

119879minus2)minus1

Φ119879minus2

119886119879minus2

(Σ1

119879minus2)minus1

119868

(32)



119906lowast

119879minus2=

(Σ1

119879minus2)minus1

119868

119886119879minus2

119909119879minus2

+1205821

119879minus2

2(Σ1

119879minus2)minus1

(120583119879minus2


119868)

+ (Σ1

119879minus2)minus1

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868] 119897119879minus2

(33)




details)

119881lowast

119879minus2(119909119879minus2

119897119879minus2

) =1

119886119879minus2

1199092


119879minus2119902119879minus2

119909119879minus2


119909119879minus2

119897119879minus2

+ 1205731119879minus2

119897119879minus2

+ 1205732119879minus2

1198972

119879minus2


)2

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(34)


119881lowast

119905(119909119905 119897119905) =

1

119886119905

1199092

119905minus 1205821

119905119902119905119909119905minus 2120574119905119909119905119897119905

+ 1205731119905119897119905+ 12057321199051198972

119905minus

119879minus1

sum119894=119905

(1205821

119894)2

4120572119894

(35)


119905minus1and



min Var119905minus1

(119878119905) minus 120582119905minus1

119864119905minus1

(119878119905)

+ 119864119905minus1

(119881lowast

119905(119909119905 119897119905))

st 1198771015840

119905minus1119906119905minus1

= 119909119905

1198681015840119906119905minus1

= 119909119905minus1

119897119905= 119876119905minus1

119897119905minus1

119878119905= 119909119905minus 119897119905

(36)


119864119905minus1

(119881lowast

119905(119909119905 119897119905)) =

1

119886119905

119864119905minus1

(1199092

119905) minus 1205821

119905119902119905119864119905minus1

(119909119905)

minus 2120574119905119864119905minus1

(119909119905119897119905) + 1205731119905119864119905minus1

(119897119905)

+ 1205732119905119864119905minus1

(1198972

119905) minus

119879minus1

sum119894=119905

(1205821

119894)2

4120572119894

(37)


(119881lowast



min 119881119905minus1

(119909119905minus1

119897119905minus1

) = 1199061015840

119879minus1Σ1

119905minus1119906119905minus1

minus 2Φ1015840

119905minus1119906119905minus1

+ Ψ119905minus1

st 1198681015840119906119905minus1

= 119909119905minus1

(38)

where

Φ119905minus1

= [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] 119897119905minus1

+1205821

119905minus1

2120583119905minus1

Ψ119905minus1

= [Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] 1198972

119905minus1

+ [120582119905minus1

+ 1205731119905] 119864 (119876

119905minus1) 119897119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4120572119894

(39)



119871 (119906119905minus1

) = 1199061015840

119879minus1Σ1

119905minus1119906119905minus1

minus 2Φ1015840

119905minus1119906119905minus1

+ Ψ119905minus1

minus 120596119905(119909119905minus1

minus 1198681015840119906119905minus1

)

(40)


2Σ1

119905minus1119906119905minus1


+ 120596119905119868 = 0

119909119905minus1

minus 1198681015840119906119905minus1

= 0

(41)

Thus we have

119906lowast

119905minus1= (Σ1

119905minus1)minus1

Φ119905minus1

minus1

2120596119905(Σ1

119905minus1)minus1

119868 (42)

1198681015840119906lowast

119905minus1= 1198681015840(Σ1

119905minus1)minus1

Φ119905minus1

minus1

21205961199051198681015840(Σ1

119905minus1)minus1

119868

= 1198681015840(Σ1

119905minus1)minus1

Φ119905minus1

minus1

2120596119905119886119905minus1

= 119909119905minus1

(43)

which implies

120596119905= 2

1198681015840(Σ1

119905minus1)minus1

Φ119905minus1

119886119905minus1

minus 21

119886119905minus1

119909119905minus1

(44)


119906lowast

119905minus1=

(Σ1

119905minus1)minus1

119868

119886119905minus1

119909119905minus1

minus1198681015840(Σ1

119905minus1)minus1

Φ119905minus1

119886119905minus1

(Σ1

119905minus1)minus1

119868

+ (Σ1

119905minus1)minus1

Φ119905minus1

=(Σ1

119905minus1)minus1

119868

119886119905minus1

119909119905minus1

+1205821

119905minus1

2(Σ1

119905minus1)minus1

(120583119905minus1


119868)

+ (Σ1

119905minus1)minus1

[Σ0


119868 + 120574119905119864 (119876119905minus1

119877119905minus1

)] 119897119905minus1

(45)




119881lowast

119905minus1(119909119905minus1

119897119905minus1

) =1

119886119905minus1

1199092


119905minus1119902119905minus1

119909119905minus1


119909119905minus1

119897119905minus1

+ 1205731119905minus1

119897119905minus1

+ 1205732119905minus1

1198972

119905minus1minus

119879minus1

sum119894=119905minus1

(1205821

119894)2

4120572119894





119906lowast

119905=

(Σ1

119905)minus1

119868

119886119905

119909119905+

1205821

119905

2(Σ1

119905)minus1

(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1

(47)





119906lowast

119905=

(Σ1

119905)minus1

119868

119886119905

119909119905+

1205821

119905

2(Σ1

119905)minus1

(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1

(48)


119905= 1198681015840(Σ1

119905)minus1Σ0

119905=

0 120574119905= 119889119905119886119905= 0 and 120574


119906lowast

119905=

(Σ1

119905)minus1

119868

119886119905

119909119905+

1205821

119905

2(Σ1

119905)minus1

(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1

(49)




(Σ1

119905)minus1

(Σ0

119905minus 120574119905119868 + 120574119905+1

119864 (119876119905119877119905)) 119897119905 (50)



119905


min119906119905

Var119905(119878119905+1

) minus 120582119905119864119905(119878119905+1

)

st 1198771015840

119905119906119905= 119909119905+1

1198681015840119906119905= 119909119905

119897119905+1

= 119876119905119897119905

119878119905+1

= 119909119905+1

minus 119897119905+1

(51)


119906119898119910

119905=

(Σ119905)minus1

119868

119886119905

119909119905+

120582119905

2(Σ119905)minus1

(120583119905minus 119902119905119868)

+ (Σ119905)minus1

(Σ0

119905minus 120574119905119868) 119897119905 119905 = 0 119879 minus 1

(52)


119905119886119905


119905rarr 0


119906lowast

119905=

(Σ1

119905)minus1

119868

119886119905

119909119905+ (Σ1

119905)minus1

(Σ0

119905minus 120574119905119868 + 120574119905+1

119864 (119876119905119877119905)) 119897119905

119905 = 0 119879 minus 1

119906119898119910

119905=

(Σ119905)minus1

119868

119886119905

119909119905+ (Σ119905)minus1

(Σ0

119905minus 120574119905119868) 119897119905 119905 = 0 119879 minus 1

(53)







119909119905= 1198751015840

119905minus1119906119905minus1

+ 119904119905minus1

119909119905minus1

(54)



0(1198781) + 1198640(1205881(1198782) + 1198641(1205882(1198783) + sdot sdot sdot

+119864119879minus3

(120588119879minus2

(119878119879minus1

) + 119864119879minus2

(120588119879minus1

(119878119879))) ))

st 119909119905+1

= 1198751015840

119905119906119905+ 119904119905119909119905

119897119905+1

= 119876119905119897119905

119878119905+1

= 119909119905+1

minus 119897119905+1

119905 = 0 1 119879 minus 1

(55)



119906lowast

119905= Σminus1

119905120593119905119897119905+

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (56)

where 120593119905= 119864(119875

119905119876119905) minus 119864(119875

119905)119864(119876119905) and 120598

119905= 119864(119875

119905)




119879minus1= 120582119879minus1

1205821

119905= 120582119905+ 1205821

119905+1119904119905+1

1205731119879minus1

= 120582119879minus1

[119864 (119876119879minus1

) minus 1205981015840


)minus1

120593119879minus1

] 1205731119905

= (120582119905+ 1205731119905+1

) 119864(119876119905) minus 1205821

1199051205981015840

119905(Σ119905)minus1120593119905

1205732119879minus1

= Var(119876119879minus1

) minus 1205931015840


)minus1

120593119879minus1

1205732119905

= Var(119876119905) + 1205732119905+1

119864(1198762

119905) minus 1205931015840

119905(Σ119905)minus1120593119905




min119906119879minus1

Var119879minus1

(119878119879) minus 120582119879minus1

119864119879minus1

(119878119879)

st 119909119879

= 119904119879minus1

119909119879minus1

+ 1198751015840

119879minus1119906119879minus1

119897119879

= 119876119879minus1

119897119879minus1

119878119879= 119909119879minus 119897119879

(57)


min119906119879minus1

119881119879minus1

(119909119879minus1

119897119879minus1

) (58)

where

119881119879minus1

(119909119879minus1

119897119879minus1

) = 1199061015840


119906119879minus1

minus 2(1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840

119879minus1)119906119879minus1

+ Var (119876119879minus1

) 1198972


119904119879minus1

119909119879minus1

+ 120582119879minus1

119864 (119876119879minus1

) 119897119879minus1

(59)


119889 (119881119879minus1

(119909119879minus1

119897119879minus1

))

119889 (119906119879minus1

)= 2Σ119879minus1

119906119879minus1

minus 2(120593119879minus1

119897119879minus1

+120582119879minus1

2120598119879minus1

)

= 0

(60)

which implies

119906lowast

119879minus1= (Σ119879minus1

)minus1

[120593119879minus1

119897119879minus1

+120582119879minus1

2120598119879minus1

] (61)




119881lowast

119879minus1(119909119879minus1

119897119879minus1

) = 1205732119879minus1

1198972

119879minus1+ 1205731119879minus1

119897119879minus1


119904119879minus1

119909119879minus1


)2

4119872119879minus1

(62)


= 1205981015840


119879minus1120598119879minus1




min Var119879minus2

(119878119879minus1

) minus 120582119879minus2

119864119879minus2

(119878119879minus1

)

+ 119864119879minus2

(119881lowast

119879minus1(119909119879minus1

119897119879minus1

))

st 119909119879minus1

= 119904119879minus2

119909119879minus2

+ 1198751015840

119879minus2119906119879minus2

119897119879minus1

= 119876119879minus2

119897119879minus2

119878119879minus1

= 119909119879minus1


(63)


119864119879minus2

(119881lowast

119879minus1(119909119879minus1

119897119879minus1

)) = 1205732119879minus1

119864 (1198762

119879minus2) 1198972

119879minus2

+ 1205731119879minus1

119864 (119876119879minus2

) 119897119879minus2


119904119879minus1

119904119879minus2

119909119879minus2


119904119879minus1

1205981015840

119879minus2119906119879minus2


)2

4119872119879minus1

(64)


(119881lowast

119879minus1(119909119879minus1

119897119879minus1


min119906119879minus2

119881119879minus2

(119909119879minus2

119897119879minus2

) (65)


(119909119879minus2

119897119879minus2

) = 1199061015840


119906119879minus2

minus 2(1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840

119879minus2)119906119879minus2

+ [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

(66)


119889 (119881119879minus2

(119909119879minus2

119897119879minus2

))

119889 (119906119879minus2

)= 2Σ119879minus2

119906119879minus2

minus 2(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= 0

(67)


Thus

119906lowast

119879minus2= (Σ119879minus2

)minus1

(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= (Σ119879minus2

)minus1

120593119879minus2

119897119879minus2

+1205821

119879minus2

2(Σ119879minus2

)minus1

120598119879minus2

(68)




119881lowast

119879minus2(119909119879minus2

119897119879minus2

) = 1205732119879minus2

1198972

119879minus2+ 1205731119879minus2

119897119879minus2

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

minus(1205821

119879minus2)2

4119872119879minus2

(69)


= 1205981015840


119879minus2120598119879minus2


both (56) and

119881lowast

119905(119909119905 119897119905) = 12057321199051198972

119905+ 1205731119905119897119905minus 1205821

119905119904119905119909119905minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

(70)

hold where 119872119894= 1205981015840

119894Σminus1



119905minus1and



min Var119905minus1

(119878119905) minus 120582119905minus1

119864119905minus1

(119878119905) + 119864119905minus1

(119881lowast

119905(119909119905 119897119905))

st 119909119905= 119904119905minus1

119909119905minus1

+ 1198751015840

119905minus1119906119905minus1

119897119905= 119876119905minus1

119897119905minus1

119878119905= 119909119905minus 119897119905

(71)


119864119905minus1

(119881lowast

119905(119909119905 119897119905)) = 120573

2119905119864 (1198762

119905minus1) 1198972

119905minus1+ 1205731119905119864 (119876119905minus1

) 119897119905

minus 1205821

119905119904119905119904119905minus1

119909119905minus1

minus 1205821

1199051199041199051205981015840

119905minus1119906119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

(72)


(119881lowast



119881119905minus1

(119909119905minus1

119897119905minus1

) (73)

where

119881119905minus1

(119909119905minus1

119897119905minus1

) = 1199061015840


119906119905minus1

minus 2(1205931015840

119905minus1119897119905minus1

+1205821

119905minus1

21205981015840

119905minus1)119906119905minus1

+ [Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] 1198972

119905minus1

+ [120582119905minus1

119864 (119876119905minus1

) + 1205731119905119864 (119876119905minus1

)] 119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

(74)


119889 (119881119905minus1

(119909119905minus1

119897119905minus1

))

119889 (119906119905minus1

)= 2Σ119905minus1

119906119905minus1

minus 2(120593119905minus1

119897119905minus1

+1205821

119905minus1

2120598119905minus1

)

= 0

(75)

which implies

119906lowast


119905minus1120593119905minus1

119897119905minus1

+1205821

119905minus1

2Σminus1

119905minus1120598119905minus1

(76)




119881lowast

119905minus1(119909119905minus1

119897119905minus1

) = 1205732119905minus1

1198972

119905minus1+ 1205731119905minus1

119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905minus1

(1205821

119894)2

4119872119894

(77)







119906lowast

119905=

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (78)






119906lowast

119905=

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (79)


120593119905= 119864 (119875

119905119876119905) minus 119864 (119875

119905) 119864 (119876

119905)

= 119864 ((119877119905minus 119904119905119868)119876119905) minus 119864 (119877

119905minus 119904119905119868) 119864 (119876

119905)

= 119864 (119877119905119876119905) minus 119864 (119877

119905) 119864 (119876

119905)

= Σ0

119905

= 0

(80)



119906lowast

119905=

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (81)




Σminus1

119905120593119905119897119905 (82)



min119906119905

Var119905(119878119905+1

) minus 120582119905119864119905(119878119905+1

)

st 119909119905+1

= 119904119905119909119905+ 1198751015840

119905119906119905

119897119905+1

= 119876119905119897119905

119878119905+1

= 119909119905+1

minus 119897119905+1

(83)


119906119898119910

119905= Σminus1

119905120593119905119897119905+

120582119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (84)




119906lowast

119905= Σminus1

119905120593119905119897119905 119905 = 0 119879 minus 1 (85)






119905


lowast

119905

0 (

11941198970+ 0903119909

0+ 3736

01071198970minus 0066119909

0+ 0203

minus05771198970+ 0163119909

0minus 3939

) (

09031199090+ 3736

minus00661199090+ 0203

01631199090minus 3939

)

1 (

minus03721198971+ 1018119909

1+ 1743

01981198971minus 0060119909

1+ 0095

minus06331198971+ 0042119909

1minus 1838

) (

10181199091+ 1743

minus00601199091+ 0095

00421199091minus 1838

)




my119905

0 (

09031199090+ 1194119897

0+ 3736

minus00661199090+ 0107119897

0+ 0203

01631199090minus 0577119897

0minus 3939

) (

10181199090+ 0449119897

0+ 1743

minus00601199090+ 0150119897

0+ 0095

00421199090minus 0599119897

0minus 1838

)

1 (

10181199091minus 0372119897

1+ 1743

minus00601199091+ 0198119897

1+ 0095

00421199091minus 0633119897

1minus 1838

) (

10181199091+ 0449119897

1+ 1743

minus00601199091+ 0150119897

1+ 0095

00421199091minus 0599119897

1minus 1838

)



119905=

(1162 1246 1228) and

Σ119905= (

00146 00187 00145

00187 00854 00104

00145 00104 00289

) (86)






119905=

05 and 120582119905







my119905

0 (

09031199090+ 1194119897

0

minus00661199090+ 0107119897

0

01631199090minus 0577119897

0

) (

10181199090+ 0449119897

0

minus00601199090+ 0150119897

0

00421199090minus 0599119897

0

)

1 (

10181199091minus 0372119897

1

minus00601199091+ 0198119897

1

00421199091minus 0633119897

1

) (

10181199091+ 0449119897

1

minus00601199091+ 0150119897

1

00421199091minus 0599119897

1

)




my119905

0 (

04441198970+ 7889

01501198970minus 0062

minus05991198970minus 3572

) (

04441198970+ 3867

01501198970minus 0031

minus05991198970minus 1751

)

1 (

04441198971+ 3867

01501198971minus 0031

minus05991198971minus 1751

) (

04441198971+ 3867

01501198971minus 0031

minus05991198971minus 1751

)




my119905

0 (

04441198970

01501198970

minus05991198970

) (

04441198970

01501198970

minus05991198970

)

1 (

04441198971

01501198971

minus05991198971

) (

04441198971

01501198971

minus05991198971

)


119905= 120583119905minus 119904119905119868 =

(0122 0206 0188)1015840 and120593

119905= Σ0

119905= (00006 00149 00050)





6 Conclusion


Appendices

A Proof of (20)


119879minus1into 119906

lowast

119879minus1

1015840Σ1


119879minus1gives

119906lowast

119879minus1

1015840

Σ1


119879minus1=

119868

119886119879minus1

119909119879minus1

+ [Σ0


119868] 119897119879minus1

+120582119879minus1

2[120583119879minus1


119868]

1015840

(Σ1

119879minus1)minus1

times 119868

119886119879minus1

119909119879minus1

+ [Σ0


119868]


+120582119879minus1

2[120583119879minus1


119868]

=1

119886119879minus1

1199092

119879minus1+ [Σ0


119868]1015840

times (Σ1

119879minus1)minus1

[Σ0


119868] 1198972

119879minus1

+ 120582119879minus1

[Σ0


119868]1015840

times (Σ1

119879minus1)minus1

[120583119879minus1


119868] 119897119879minus1

+(120582119879minus1

)2

4[120583119879minus1


119868]1015840

times (Σ1

119879minus1)minus1

[120583119879minus1


119868]

=1

119886119879minus1

1199092

119879minus1+ [(Σ

0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0

119879minus1


1205742

119879minus1] 1198972

119879minus1

+ [(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1


120574119879minus1

]


119897119879minus1

+(120582119879minus1

)2

4120572119879minus1

(A1)



0

119879minus1)1015840

119897119879minus1

+

(120582119879minus1

2)1205831015840



2 [(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840


119879minus1

=2

119886119879minus1

[(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1

119868119909119879minus1

+ 2 [(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1


times [Σ0


119868] 119897119879minus1

+ 120582119879minus1

[(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1



119868]

= 2120574119879minus1

119897119879minus1

119909119879minus1

+ 119902119879minus1

120582119879minus1

119909119879minus1

+ 2 [(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0


119889119879minus1

] 1198972

119879minus1

+ 2120582119879minus1

[(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1


120574119879minus1

] 119897119879minus1

+(120582119879minus1

)2

2120572119879minus1

(A2)

Thus119881lowast

119879minus1(119909119879minus1

119897119879minus1

)

=1

119886119879minus1

1199092


119902119879minus1

119909119879minus1


119909119879minus1

119897119879minus1

+ 120582119879minus1

[119864 (119876119879minus1

) minus (Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1

+119887119879minus1

120574119879minus1

] 119897119879minus1

+ [Var (119876119879minus1

) minus (Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0

119879minus1

+120574119879minus1

119889119879minus1

] 1198972

119879minus1minus

(120582119879minus1

)2

4120572119879minus1

=1

119886119879minus1

1199092


119902119879minus1

119909119879minus1


119909119879minus1

119897119879minus1

+ 1205731119879minus1

119897119879minus1

+ 1205732119879minus1

1198972

119879minus1minus

(120582119879minus1

)2

4120572119879minus1

(A3)


1198681015840

119886119879minus2

(Σ1

119879minus2)minus1

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

=119889119879minus2

119886119879minus2

+ 120574119879minus1

1198681015840(Σ1

119879minus2)minus1

119864 (119876119879minus2

119877119879minus2

)

119886119879minus2


= 120574119879minus2

+ 120574119879minus1

1198681015840(Σ1

119879minus2)minus1

119864 (119876119879minus2

119877119879minus2

)

119886119879minus2


= 120574119879minus2


= 0

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1

[120583119879minus2


119868] =119887119879minus2

119886119879minus2


= 0

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

119868

= 119886119879minus2

120574119879minus2

+ 120574119879minus1

119864 (119876119879minus2

1198771015840

119879minus2) (Σ1

119879minus2)minus1

119868

119886119879minus2

= 119886119879minus2

120574119879minus2

(B1)


119879minus2

1015840Σ1




and 119897119879minus2

119909119879minus2

in 119906lowast

119879minus2

1015840Σ1



119879minus2is

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1 119868

119886119879minus2

=1

119886119879minus2

(B2)


is

1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

= 1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

minus 1205821

119879minus2120574119879minus2

1198681015840(Σ1

119879minus2)minus1

[120583119879minus2


119868]

= 1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

(B3)


is

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]1015840

sdot (Σ1

119879minus2)minus1

sdot [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

= [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]


1205742

119879minus2

(B4)


(1205821

119879minus2)2

4[120583119879minus2


119868]1015840

(Σ1

119879minus2)minus1

[120583119879minus2


119868]

=(1205821

119879minus2)2

4120572119879minus2

(B5)




119879minus2


is

1205821

119879minus2

2

1205831015840

119879minus2(Σ1

119879minus2)minus1

119868

119886119879minus2

=1205821

119879minus2119902119879minus2

2 (B6)



119909119879minus2

is 120574119879minus2


is

1205821

119879minus2

2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

(120583119879minus2


119868)

+1205821

119879minus2

21205831015840

119879minus2(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

(B7)


is

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

= [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)] minus 119886119879minus2

1205742

119879minus2

(B8)


(1205821

119879minus2

2)

2

1205831015840

119879minus2(Σ1

119879minus2)minus1

(120583119879minus2


119868) = (1205821

119879minus2

2)

2

120572119879minus2

(B9)


119879minus2(119909119879minus2

119897119879minus2


119879minus2is

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1 119868

119886119879minus2

=1

119886119879minus2

(B10)



119902119879minus2


119909119879minus2



is

[Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] + 119886

119879minus21205742

119879minus2

minus [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]

= 1205732119879minus2

(B11)


is

minus 1205821

119879minus21205831015840

119879minus2(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)]

= 1205731119879minus2

(B12)


minus1205822

119879minus1

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(B13)

Thus

119881lowast

119879minus2(119909119879minus2

119897119879minus2

) =1

119886119879minus2

1199092


119879minus2119902119879minus2

119909119879minus2


119909119879minus2

119897119879minus2

+ 1205731119879minus2

119897119879minus2

+ 1205732119879minus2

1198972

119879minus2minus

1205822

119879minus1

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(B14)


1198681015840

119886119905minus1

(Σ1

119905minus1)minus1

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

=119889119905minus1

119886119905minus1

+ 120574119905

1198681015840(Σ1

119905minus1)minus1

119864 (119876119905minus1

119877119905minus1

)

119886119905minus1


= 120574119905minus1

+ 120574119905

1198681015840(Σ1

119905minus1)minus1

119864 (119876119905minus1

119877119905minus1

)

119886119905minus1


= 120574119905minus1


= 0

1198681015840

119886119905minus1

(Σ1

119905minus1)minus1

[120583119905minus1


119868] =119887119905minus1

119886119905minus1


= 0

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

119868

= 119886119905minus1

120574119905minus1

+ 120574119905

119864 (119876119905minus1

1198771015840

119905minus1) (Σ1

119905minus1)minus1

119868

119886119905minus1

= 119886119905minus1

120574119905minus1

(C1)


119905minus1

1015840Σ1




and 119897119905minus1

119909119905minus1


is

1198681015840

119886119905minus1

(Σ1

119905minus1)minus1 119868

119886119905minus1

=1

119886119905minus1

(C2)


is

1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

= 1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

minus 1205821

119905minus1120574119905minus1

1198681015840(Σ1

119905minus1)minus1

[120583119905minus1


119868]


= 1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

(C3)


is

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]1015840

sdot (Σ1

119905minus1)minus1

sdot [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

= [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] minus 119886119905minus1

1205742

119905minus1

(C4)


(1205821

119905minus1)2

4[120583119905minus1


119868]1015840

(Σ1

119905minus1)minus1

[120583119905minus1


119868] =(1205821

119905minus1)2

4120572119905minus1

(C5)




119905minus1is


is

1205821

119905minus1

2

1205831015840

119905minus1(Σ1

119905minus1)minus1

119868

119886119905minus1

=1205821

119905minus1119902119905minus1

2 (C6)


119909119905minus1

is 120574119905minus1


is

1205821

119905minus1

2[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1



119868) +1205821

119905minus1

21205831015840

119905minus1(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

(C7)


is

[Σ0

119905minus1+ 120574119905119864(119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

= [Σ0

119905minus1+ 120574119905119864(119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] minus 119886119905minus1

1205742

119905minus1

(C8)


(1205821

119905minus1

2)

2

120583119905minus1

(Σ1

119905minus1)minus1

(120583119905minus1


119868) = (1205821

119905minus1

2)

2

120572119905minus1

(C9)


119905minus1(119909119905minus1

119897119905minus1


119905minus1is 1119886



1

119905minus1119902119905minus1


119905minus1119909119905minus1



is

[Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] + 119886

119905minus11205742

119905minus1

minus [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]

= 1205732119905minus1

(C10)


is

minus 1205821

119905minus11205831015840

119905minus1(Σ1

119905minus1)minus1

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

+ (120582119905minus1

+ 1205731119905) 119864 (119876

119905minus1) = 1205731119905minus1

(C11)


(1205821

119894)2120572119894

Thus

119881lowast

119905minus1(119909119905minus1

119897119905minus1

) =1

119886119905minus1

1199092


119905minus1119902119905minus1

119909119905minus1


119909119905minus1

119897119905minus1

+ 1205731119905minus1

119897119905minus1

+ 1205732119905minus1

1198972

119905minus1minus

1

4

119879minus1

sum119894=119905minus1

(1205821

119894)2

120572119894

(C12)

D Proof of (62)


119879minus1into 119906

lowast

119879minus1

1015840Σ119879minus1

119906lowast


119906lowast

119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1= (1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840

119879minus1)


)minus1

(120593119879minus1

119897119879minus1

+120582119879minus1

2120598119879minus1

)

= 1205931015840


)minus1

120593119879minus1

1198972

119879minus1+ 120582119879minus1

1205981015840

119879minus1


)minus1

120593119879minus1

119897119879minus1

+(120582119879minus1

)2

4119872119879minus1

2 (1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840



119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1

(D1)

Thus

119881lowast

119879minus1(119909119879minus1

119897119879minus1


119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1+ Var (119876

119879minus1) 1198972

119879minus1


119904119879minus1

119909119879minus1

+ 120582119879minus1

119864 (119876119879minus1

) 119897119879minus1

= [Var (119876119879minus1

) minus 1205931015840


)minus1

120593119879minus1

] 1198972

119879minus1

+ 120582119879minus1

[119864 (119876119879minus1

) minus 1205981015840


)minus1

120593119879minus1

]



119904119879minus1

119909119879minus1


)2

4119872119879minus1

= 1205732119879minus1

1198972

119879minus1+ 1205731119879minus1

119897119879minus1


119904119879minus1

119909119879minus1


)2

4119872119879minus1

(D2)



lowast


119906lowast

119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2= (1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840

119879minus2)


)minus1

(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= 1205931015840


)minus1

120593119879minus2

1198972

119879minus2+ 1205821

119879minus21205981015840

119879minus2


)minus1

120593119879minus2

119897119879minus2

+(1205821

119879minus2)2

4119872119879minus2

2 1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840



119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2

(E1)Thus

119881lowast

119879minus2(119909119879minus2

119897119879minus2


119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2

+ [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

= 1205732119879minus2

1198972

119879minus2+ 1205731119879minus2

119897119879minus2

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

minus(1205821

119879minus2)2

4119872119879minus2

(E2)where1205731119879minus2

= (120582119879minus2

+ 1205731119879minus1

) 119864 (119876119879minus2

) minus 1205821

119879minus21205981015840


)minus1

120593119879minus2

1205732119879minus2

= Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2) minus 1205931015840


)minus1

120593119879minus2

(E3)


lowast

119905minus1that

119906lowast

119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1= (1205931015840

119879minus1119897119905minus1

+1205821

119905minus1

21205981015840

119879minus1)


)minus1

(120593119905minus1

119897119905minus1

+1205821

119905minus1

2120598119905minus1

)

= 1205931015840


)minus1

120593119905minus1

1198972

119905minus1+ 1205821

119905minus11205981015840

119879minus1


)minus1

120593119905minus1

119897119905minus1

+(1205821

119905minus1)2

4119872119905minus1

2 (1205931015840

119879minus1119897119905minus1

+1205821

119905minus1

21205981015840



119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1

(F1)

Thus

119881lowast

119905minus1(119909119905minus1

119897119905minus1


119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1

+ [Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] 1198972

119905minus1

+ [120582119905minus1

119864 (119876119905minus1

) + 1205731119905119864 (119876119905minus1

)] 119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

= 1205732119905minus1

1198972

119905minus1+ 1205731119905minus1

119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905minus1

(1205821

119894)2

4119872119894

(F2)

where

1205731119905minus1

= (120582119905minus1

+ 1205731119905) 119864 (119876

119905minus1) minus 1205821

119905minus11205981015840


)minus1

120593119905minus1

1205732119905minus1

= Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1) minus 1205931015840


)minus1

120593119905minus1

(F3)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119881lowast

119879minus1(119909119879minus1

119897119879minus1

) =1

119886119879minus1

1199092


119902119879minus1

119909119879minus1


119909119879minus1

119897119879minus1

+ 1205731119879minus1

119897119879minus1

+ 1205732119879minus1

1198972

119879minus1minus

(120582119879minus1

)2

4120572119879minus1

(20)




min Var119879minus2

(119878119879minus1

) minus 120582119879minus2

119864119879minus2

(119878119879minus1

)

+ 119864119879minus2

(119881lowast

119879minus1(119909119879minus1

119897119879minus1

))

st 1198771015840

119879minus2119906119879minus2

= 119909119879minus1

1198681015840119906119879minus2

= 119909119879minus2

119897119879minus1

= 119876119879minus2

119897119879minus2

119878119879minus1

= 119909119879minus1


(21)


119864119879minus2

(119881lowast

119879minus1(119909119879minus1

119897119879minus1

))

=1

119886119879minus1

119864119879minus2

(1199092


119902119879minus1

119864119879minus2

(119909119879minus1

)


119864119879minus2

(119909119879minus1

119897119879minus1

) + 1205731119879minus1

119864119879minus2

(119897119879minus1

)

+ 1205732119879minus1

119864119879minus2

(1198972

119879minus1) minus

(120582119879minus1

)2

4120572119879minus1

(22)


(119881lowast

119879minus1(119909119879minus1

119897119879minus1


min 119881119879minus2

(119909119879minus2

119897119879minus2

)

st 1198681015840119906119879minus2

= 119909119879minus2

(23)

where

119881119879minus2

(119909119879minus2

119897119879minus2

) = 1199061015840

119879minus2Σ1

119879minus2119906119879minus2

minus 2 [(Σ0

119879minus2)1015840

+ 120574119879minus1

119864 (119876119879minus2

1198771015840

119879minus2)] 119897119879minus2

+1205821

119879minus2

21205831015840

119879minus2119906119879minus2

+ [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1


)2

4120572119879minus1

(24)



119871 (119906119879minus2

) = 1199061015840

119879minus2Σ1

119879minus2119906119879minus2

minus 2Φ1015840

119879minus2119906119879minus2

+ Ψ119879minus2


(119909119879minus2

minus 1198681015840119906119879minus2

)

(25)

where

Φ119879minus2

= [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)] 119897119879minus2

+1205821

119879minus2

2120583119879minus2

Ψ119879minus2

= [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1


)2

4120572119879minus1

(26)


2Σ1

119879minus2119906119879minus2


+ 120596119879minus1

119868 = 0 (27)

119909119879minus2

minus 1198681015840119906119879minus2

= 0 (28)

From (27) we have

119906lowast

119879minus2= (Σ1

119879minus2)minus1

Φ119879minus2

minus1

2120596119879minus1

(Σ1

119879minus2)minus1

119868 (29)

1198681015840119906lowast

119879minus2= 1198681015840(Σ1

119879minus2)minus1

Φ119879minus2

minus1

2120596119879minus1

1198681015840(Σ1

119879minus2)minus1

119868

= 1198681015840(Σ1

119879minus2)minus1

Φ119879minus2

minus1

2120596119879minus1

119886119879minus2

= 119909119879minus2

(30)

which implies

120596119879minus1

= 21198681015840(Σ1

119879minus2)minus1

Φ119879minus2

119886119879minus2

minus 21

119886119879minus2

119909119879minus2

(31)


into (29) we get

119906lowast

119879minus2=

(Σ1

119879minus2)minus1

119868

119886119879minus2

119909119879minus2

+ (Σ1

119879minus2)minus1

Φ119879minus2

minus1198681015840(Σ1

119879minus2)minus1

Φ119879minus2

119886119879minus2

(Σ1

119879minus2)minus1

119868

(32)



119906lowast

119879minus2=

(Σ1

119879minus2)minus1

119868

119886119879minus2

119909119879minus2

+1205821

119879minus2

2(Σ1

119879minus2)minus1

(120583119879minus2


119868)

+ (Σ1

119879minus2)minus1

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868] 119897119879minus2

(33)




details)

119881lowast

119879minus2(119909119879minus2

119897119879minus2

) =1

119886119879minus2

1199092


119879minus2119902119879minus2

119909119879minus2


119909119879minus2

119897119879minus2

+ 1205731119879minus2

119897119879minus2

+ 1205732119879minus2

1198972

119879minus2


)2

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(34)


119881lowast

119905(119909119905 119897119905) =

1

119886119905

1199092

119905minus 1205821

119905119902119905119909119905minus 2120574119905119909119905119897119905

+ 1205731119905119897119905+ 12057321199051198972

119905minus

119879minus1

sum119894=119905

(1205821

119894)2

4120572119894

(35)


119905minus1and



min Var119905minus1

(119878119905) minus 120582119905minus1

119864119905minus1

(119878119905)

+ 119864119905minus1

(119881lowast

119905(119909119905 119897119905))

st 1198771015840

119905minus1119906119905minus1

= 119909119905

1198681015840119906119905minus1

= 119909119905minus1

119897119905= 119876119905minus1

119897119905minus1

119878119905= 119909119905minus 119897119905

(36)


119864119905minus1

(119881lowast

119905(119909119905 119897119905)) =

1

119886119905

119864119905minus1

(1199092

119905) minus 1205821

119905119902119905119864119905minus1

(119909119905)

minus 2120574119905119864119905minus1

(119909119905119897119905) + 1205731119905119864119905minus1

(119897119905)

+ 1205732119905119864119905minus1

(1198972

119905) minus

119879minus1

sum119894=119905

(1205821

119894)2

4120572119894

(37)


(119881lowast



min 119881119905minus1

(119909119905minus1

119897119905minus1

) = 1199061015840

119879minus1Σ1

119905minus1119906119905minus1

minus 2Φ1015840

119905minus1119906119905minus1

+ Ψ119905minus1

st 1198681015840119906119905minus1

= 119909119905minus1

(38)

where

Φ119905minus1

= [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] 119897119905minus1

+1205821

119905minus1

2120583119905minus1

Ψ119905minus1

= [Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] 1198972

119905minus1

+ [120582119905minus1

+ 1205731119905] 119864 (119876

119905minus1) 119897119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4120572119894

(39)



119871 (119906119905minus1

) = 1199061015840

119879minus1Σ1

119905minus1119906119905minus1

minus 2Φ1015840

119905minus1119906119905minus1

+ Ψ119905minus1

minus 120596119905(119909119905minus1

minus 1198681015840119906119905minus1

)

(40)


2Σ1

119905minus1119906119905minus1


+ 120596119905119868 = 0

119909119905minus1

minus 1198681015840119906119905minus1

= 0

(41)

Thus we have

119906lowast

119905minus1= (Σ1

119905minus1)minus1

Φ119905minus1

minus1

2120596119905(Σ1

119905minus1)minus1

119868 (42)

1198681015840119906lowast

119905minus1= 1198681015840(Σ1

119905minus1)minus1

Φ119905minus1

minus1

21205961199051198681015840(Σ1

119905minus1)minus1

119868

= 1198681015840(Σ1

119905minus1)minus1

Φ119905minus1

minus1

2120596119905119886119905minus1

= 119909119905minus1

(43)

which implies

120596119905= 2

1198681015840(Σ1

119905minus1)minus1

Φ119905minus1

119886119905minus1

minus 21

119886119905minus1

119909119905minus1

(44)


119906lowast

119905minus1=

(Σ1

119905minus1)minus1

119868

119886119905minus1

119909119905minus1

minus1198681015840(Σ1

119905minus1)minus1

Φ119905minus1

119886119905minus1

(Σ1

119905minus1)minus1

119868

+ (Σ1

119905minus1)minus1

Φ119905minus1

=(Σ1

119905minus1)minus1

119868

119886119905minus1

119909119905minus1

+1205821

119905minus1

2(Σ1

119905minus1)minus1

(120583119905minus1


119868)

+ (Σ1

119905minus1)minus1

[Σ0


119868 + 120574119905119864 (119876119905minus1

119877119905minus1

)] 119897119905minus1

(45)




119881lowast

119905minus1(119909119905minus1

119897119905minus1

) =1

119886119905minus1

1199092


119905minus1119902119905minus1

119909119905minus1


119909119905minus1

119897119905minus1

+ 1205731119905minus1

119897119905minus1

+ 1205732119905minus1

1198972

119905minus1minus

119879minus1

sum119894=119905minus1

(1205821

119894)2

4120572119894





119906lowast

119905=

(Σ1

119905)minus1

119868

119886119905

119909119905+

1205821

119905

2(Σ1

119905)minus1

(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1

(47)





119906lowast

119905=

(Σ1

119905)minus1

119868

119886119905

119909119905+

1205821

119905

2(Σ1

119905)minus1

(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1

(48)


119905= 1198681015840(Σ1

119905)minus1Σ0

119905=

0 120574119905= 119889119905119886119905= 0 and 120574


119906lowast

119905=

(Σ1

119905)minus1

119868

119886119905

119909119905+

1205821

119905

2(Σ1

119905)minus1

(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1

(49)




(Σ1

119905)minus1

(Σ0

119905minus 120574119905119868 + 120574119905+1

119864 (119876119905119877119905)) 119897119905 (50)



119905


min119906119905

Var119905(119878119905+1

) minus 120582119905119864119905(119878119905+1

)

st 1198771015840

119905119906119905= 119909119905+1

1198681015840119906119905= 119909119905

119897119905+1

= 119876119905119897119905

119878119905+1

= 119909119905+1

minus 119897119905+1

(51)


119906119898119910

119905=

(Σ119905)minus1

119868

119886119905

119909119905+

120582119905

2(Σ119905)minus1

(120583119905minus 119902119905119868)

+ (Σ119905)minus1

(Σ0

119905minus 120574119905119868) 119897119905 119905 = 0 119879 minus 1

(52)


119905119886119905


119905rarr 0


119906lowast

119905=

(Σ1

119905)minus1

119868

119886119905

119909119905+ (Σ1

119905)minus1

(Σ0

119905minus 120574119905119868 + 120574119905+1

119864 (119876119905119877119905)) 119897119905

119905 = 0 119879 minus 1

119906119898119910

119905=

(Σ119905)minus1

119868

119886119905

119909119905+ (Σ119905)minus1

(Σ0

119905minus 120574119905119868) 119897119905 119905 = 0 119879 minus 1

(53)







119909119905= 1198751015840

119905minus1119906119905minus1

+ 119904119905minus1

119909119905minus1

(54)



0(1198781) + 1198640(1205881(1198782) + 1198641(1205882(1198783) + sdot sdot sdot

+119864119879minus3

(120588119879minus2

(119878119879minus1

) + 119864119879minus2

(120588119879minus1

(119878119879))) ))

st 119909119905+1

= 1198751015840

119905119906119905+ 119904119905119909119905

119897119905+1

= 119876119905119897119905

119878119905+1

= 119909119905+1

minus 119897119905+1

119905 = 0 1 119879 minus 1

(55)



119906lowast

119905= Σminus1

119905120593119905119897119905+

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (56)

where 120593119905= 119864(119875

119905119876119905) minus 119864(119875

119905)119864(119876119905) and 120598

119905= 119864(119875

119905)




119879minus1= 120582119879minus1

1205821

119905= 120582119905+ 1205821

119905+1119904119905+1

1205731119879minus1

= 120582119879minus1

[119864 (119876119879minus1

) minus 1205981015840


)minus1

120593119879minus1

] 1205731119905

= (120582119905+ 1205731119905+1

) 119864(119876119905) minus 1205821

1199051205981015840

119905(Σ119905)minus1120593119905

1205732119879minus1

= Var(119876119879minus1

) minus 1205931015840


)minus1

120593119879minus1

1205732119905

= Var(119876119905) + 1205732119905+1

119864(1198762

119905) minus 1205931015840

119905(Σ119905)minus1120593119905




min119906119879minus1

Var119879minus1

(119878119879) minus 120582119879minus1

119864119879minus1

(119878119879)

st 119909119879

= 119904119879minus1

119909119879minus1

+ 1198751015840

119879minus1119906119879minus1

119897119879

= 119876119879minus1

119897119879minus1

119878119879= 119909119879minus 119897119879

(57)


min119906119879minus1

119881119879minus1

(119909119879minus1

119897119879minus1

) (58)

where

119881119879minus1

(119909119879minus1

119897119879minus1

) = 1199061015840


119906119879minus1

minus 2(1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840

119879minus1)119906119879minus1

+ Var (119876119879minus1

) 1198972


119904119879minus1

119909119879minus1

+ 120582119879minus1

119864 (119876119879minus1

) 119897119879minus1

(59)


119889 (119881119879minus1

(119909119879minus1

119897119879minus1

))

119889 (119906119879minus1

)= 2Σ119879minus1

119906119879minus1

minus 2(120593119879minus1

119897119879minus1

+120582119879minus1

2120598119879minus1

)

= 0

(60)

which implies

119906lowast

119879minus1= (Σ119879minus1

)minus1

[120593119879minus1

119897119879minus1

+120582119879minus1

2120598119879minus1

] (61)




119881lowast

119879minus1(119909119879minus1

119897119879minus1

) = 1205732119879minus1

1198972

119879minus1+ 1205731119879minus1

119897119879minus1


119904119879minus1

119909119879minus1


)2

4119872119879minus1

(62)


= 1205981015840


119879minus1120598119879minus1




min Var119879minus2

(119878119879minus1

) minus 120582119879minus2

119864119879minus2

(119878119879minus1

)

+ 119864119879minus2

(119881lowast

119879minus1(119909119879minus1

119897119879minus1

))

st 119909119879minus1

= 119904119879minus2

119909119879minus2

+ 1198751015840

119879minus2119906119879minus2

119897119879minus1

= 119876119879minus2

119897119879minus2

119878119879minus1

= 119909119879minus1


(63)


119864119879minus2

(119881lowast

119879minus1(119909119879minus1

119897119879minus1

)) = 1205732119879minus1

119864 (1198762

119879minus2) 1198972

119879minus2

+ 1205731119879minus1

119864 (119876119879minus2

) 119897119879minus2


119904119879minus1

119904119879minus2

119909119879minus2


119904119879minus1

1205981015840

119879minus2119906119879minus2


)2

4119872119879minus1

(64)


(119881lowast

119879minus1(119909119879minus1

119897119879minus1


min119906119879minus2

119881119879minus2

(119909119879minus2

119897119879minus2

) (65)


(119909119879minus2

119897119879minus2

) = 1199061015840


119906119879minus2

minus 2(1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840

119879minus2)119906119879minus2

+ [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

(66)


119889 (119881119879minus2

(119909119879minus2

119897119879minus2

))

119889 (119906119879minus2

)= 2Σ119879minus2

119906119879minus2

minus 2(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= 0

(67)


Thus

119906lowast

119879minus2= (Σ119879minus2

)minus1

(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= (Σ119879minus2

)minus1

120593119879minus2

119897119879minus2

+1205821

119879minus2

2(Σ119879minus2

)minus1

120598119879minus2

(68)




119881lowast

119879minus2(119909119879minus2

119897119879minus2

) = 1205732119879minus2

1198972

119879minus2+ 1205731119879minus2

119897119879minus2

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

minus(1205821

119879minus2)2

4119872119879minus2

(69)


= 1205981015840


119879minus2120598119879minus2


both (56) and

119881lowast

119905(119909119905 119897119905) = 12057321199051198972

119905+ 1205731119905119897119905minus 1205821

119905119904119905119909119905minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

(70)

hold where 119872119894= 1205981015840

119894Σminus1



119905minus1and



min Var119905minus1

(119878119905) minus 120582119905minus1

119864119905minus1

(119878119905) + 119864119905minus1

(119881lowast

119905(119909119905 119897119905))

st 119909119905= 119904119905minus1

119909119905minus1

+ 1198751015840

119905minus1119906119905minus1

119897119905= 119876119905minus1

119897119905minus1

119878119905= 119909119905minus 119897119905

(71)


119864119905minus1

(119881lowast

119905(119909119905 119897119905)) = 120573

2119905119864 (1198762

119905minus1) 1198972

119905minus1+ 1205731119905119864 (119876119905minus1

) 119897119905

minus 1205821

119905119904119905119904119905minus1

119909119905minus1

minus 1205821

1199051199041199051205981015840

119905minus1119906119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

(72)


(119881lowast



119881119905minus1

(119909119905minus1

119897119905minus1

) (73)

where

119881119905minus1

(119909119905minus1

119897119905minus1

) = 1199061015840


119906119905minus1

minus 2(1205931015840

119905minus1119897119905minus1

+1205821

119905minus1

21205981015840

119905minus1)119906119905minus1

+ [Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] 1198972

119905minus1

+ [120582119905minus1

119864 (119876119905minus1

) + 1205731119905119864 (119876119905minus1

)] 119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

(74)


119889 (119881119905minus1

(119909119905minus1

119897119905minus1

))

119889 (119906119905minus1

)= 2Σ119905minus1

119906119905minus1

minus 2(120593119905minus1

119897119905minus1

+1205821

119905minus1

2120598119905minus1

)

= 0

(75)

which implies

119906lowast


119905minus1120593119905minus1

119897119905minus1

+1205821

119905minus1

2Σminus1

119905minus1120598119905minus1

(76)




119881lowast

119905minus1(119909119905minus1

119897119905minus1

) = 1205732119905minus1

1198972

119905minus1+ 1205731119905minus1

119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905minus1

(1205821

119894)2

4119872119894

(77)







119906lowast

119905=

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (78)






119906lowast

119905=

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (79)


120593119905= 119864 (119875

119905119876119905) minus 119864 (119875

119905) 119864 (119876

119905)

= 119864 ((119877119905minus 119904119905119868)119876119905) minus 119864 (119877

119905minus 119904119905119868) 119864 (119876

119905)

= 119864 (119877119905119876119905) minus 119864 (119877

119905) 119864 (119876

119905)

= Σ0

119905

= 0

(80)



119906lowast

119905=

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (81)




Σminus1

119905120593119905119897119905 (82)



min119906119905

Var119905(119878119905+1

) minus 120582119905119864119905(119878119905+1

)

st 119909119905+1

= 119904119905119909119905+ 1198751015840

119905119906119905

119897119905+1

= 119876119905119897119905

119878119905+1

= 119909119905+1

minus 119897119905+1

(83)


119906119898119910

119905= Σminus1

119905120593119905119897119905+

120582119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (84)




119906lowast

119905= Σminus1

119905120593119905119897119905 119905 = 0 119879 minus 1 (85)






119905


lowast

119905

0 (

11941198970+ 0903119909

0+ 3736

01071198970minus 0066119909

0+ 0203

minus05771198970+ 0163119909

0minus 3939

) (

09031199090+ 3736

minus00661199090+ 0203

01631199090minus 3939

)

1 (

minus03721198971+ 1018119909

1+ 1743

01981198971minus 0060119909

1+ 0095

minus06331198971+ 0042119909

1minus 1838

) (

10181199091+ 1743

minus00601199091+ 0095

00421199091minus 1838

)




my119905

0 (

09031199090+ 1194119897

0+ 3736

minus00661199090+ 0107119897

0+ 0203

01631199090minus 0577119897

0minus 3939

) (

10181199090+ 0449119897

0+ 1743

minus00601199090+ 0150119897

0+ 0095

00421199090minus 0599119897

0minus 1838

)

1 (

10181199091minus 0372119897

1+ 1743

minus00601199091+ 0198119897

1+ 0095

00421199091minus 0633119897

1minus 1838

) (

10181199091+ 0449119897

1+ 1743

minus00601199091+ 0150119897

1+ 0095

00421199091minus 0599119897

1minus 1838

)



119905=

(1162 1246 1228) and

Σ119905= (

00146 00187 00145

00187 00854 00104

00145 00104 00289

) (86)






119905=

05 and 120582119905







my119905

0 (

09031199090+ 1194119897

0

minus00661199090+ 0107119897

0

01631199090minus 0577119897

0

) (

10181199090+ 0449119897

0

minus00601199090+ 0150119897

0

00421199090minus 0599119897

0

)

1 (

10181199091minus 0372119897

1

minus00601199091+ 0198119897

1

00421199091minus 0633119897

1

) (

10181199091+ 0449119897

1

minus00601199091+ 0150119897

1

00421199091minus 0599119897

1

)




my119905

0 (

04441198970+ 7889

01501198970minus 0062

minus05991198970minus 3572

) (

04441198970+ 3867

01501198970minus 0031

minus05991198970minus 1751

)

1 (

04441198971+ 3867

01501198971minus 0031

minus05991198971minus 1751

) (

04441198971+ 3867

01501198971minus 0031

minus05991198971minus 1751

)




my119905

0 (

04441198970

01501198970

minus05991198970

) (

04441198970

01501198970

minus05991198970

)

1 (

04441198971

01501198971

minus05991198971

) (

04441198971

01501198971

minus05991198971

)


119905= 120583119905minus 119904119905119868 =

(0122 0206 0188)1015840 and120593

119905= Σ0

119905= (00006 00149 00050)





6 Conclusion


Appendices

A Proof of (20)


119879minus1into 119906

lowast

119879minus1

1015840Σ1


119879minus1gives

119906lowast

119879minus1

1015840

Σ1


119879minus1=

119868

119886119879minus1

119909119879minus1

+ [Σ0


119868] 119897119879minus1

+120582119879minus1

2[120583119879minus1


119868]

1015840

(Σ1

119879minus1)minus1

times 119868

119886119879minus1

119909119879minus1

+ [Σ0


119868]


+120582119879minus1

2[120583119879minus1


119868]

=1

119886119879minus1

1199092

119879minus1+ [Σ0


119868]1015840

times (Σ1

119879minus1)minus1

[Σ0


119868] 1198972

119879minus1

+ 120582119879minus1

[Σ0


119868]1015840

times (Σ1

119879minus1)minus1

[120583119879minus1


119868] 119897119879minus1

+(120582119879minus1

)2

4[120583119879minus1


119868]1015840

times (Σ1

119879minus1)minus1

[120583119879minus1


119868]

=1

119886119879minus1

1199092

119879minus1+ [(Σ

0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0

119879minus1


1205742

119879minus1] 1198972

119879minus1

+ [(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1


120574119879minus1

]


119897119879minus1

+(120582119879minus1

)2

4120572119879minus1

(A1)



0

119879minus1)1015840

119897119879minus1

+

(120582119879minus1

2)1205831015840



2 [(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840


119879minus1

=2

119886119879minus1

[(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1

119868119909119879minus1

+ 2 [(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1


times [Σ0


119868] 119897119879minus1

+ 120582119879minus1

[(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1



119868]

= 2120574119879minus1

119897119879minus1

119909119879minus1

+ 119902119879minus1

120582119879minus1

119909119879minus1

+ 2 [(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0


119889119879minus1

] 1198972

119879minus1

+ 2120582119879minus1

[(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1


120574119879minus1

] 119897119879minus1

+(120582119879minus1

)2

2120572119879minus1

(A2)

Thus119881lowast

119879minus1(119909119879minus1

119897119879minus1

)

=1

119886119879minus1

1199092


119902119879minus1

119909119879minus1


119909119879minus1

119897119879minus1

+ 120582119879minus1

[119864 (119876119879minus1

) minus (Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1

+119887119879minus1

120574119879minus1

] 119897119879minus1

+ [Var (119876119879minus1

) minus (Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0

119879minus1

+120574119879minus1

119889119879minus1

] 1198972

119879minus1minus

(120582119879minus1

)2

4120572119879minus1

=1

119886119879minus1

1199092


119902119879minus1

119909119879minus1


119909119879minus1

119897119879minus1

+ 1205731119879minus1

119897119879minus1

+ 1205732119879minus1

1198972

119879minus1minus

(120582119879minus1

)2

4120572119879minus1

(A3)


1198681015840

119886119879minus2

(Σ1

119879minus2)minus1

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

=119889119879minus2

119886119879minus2

+ 120574119879minus1

1198681015840(Σ1

119879minus2)minus1

119864 (119876119879minus2

119877119879minus2

)

119886119879minus2


= 120574119879minus2

+ 120574119879minus1

1198681015840(Σ1

119879minus2)minus1

119864 (119876119879minus2

119877119879minus2

)

119886119879minus2


= 120574119879minus2


= 0

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1

[120583119879minus2


119868] =119887119879minus2

119886119879minus2


= 0

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

119868

= 119886119879minus2

120574119879minus2

+ 120574119879minus1

119864 (119876119879minus2

1198771015840

119879minus2) (Σ1

119879minus2)minus1

119868

119886119879minus2

= 119886119879minus2

120574119879minus2

(B1)


119879minus2

1015840Σ1




and 119897119879minus2

119909119879minus2

in 119906lowast

119879minus2

1015840Σ1



119879minus2is

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1 119868

119886119879minus2

=1

119886119879minus2

(B2)


is

1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

= 1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

minus 1205821

119879minus2120574119879minus2

1198681015840(Σ1

119879minus2)minus1

[120583119879minus2


119868]

= 1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

(B3)


is

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]1015840

sdot (Σ1

119879minus2)minus1

sdot [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

= [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]


1205742

119879minus2

(B4)


(1205821

119879minus2)2

4[120583119879minus2


119868]1015840

(Σ1

119879minus2)minus1

[120583119879minus2


119868]

=(1205821

119879minus2)2

4120572119879minus2

(B5)




119879minus2


is

1205821

119879minus2

2

1205831015840

119879minus2(Σ1

119879minus2)minus1

119868

119886119879minus2

=1205821

119879minus2119902119879minus2

2 (B6)



119909119879minus2

is 120574119879minus2


is

1205821

119879minus2

2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

(120583119879minus2


119868)

+1205821

119879minus2

21205831015840

119879minus2(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

(B7)


is

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

= [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)] minus 119886119879minus2

1205742

119879minus2

(B8)


(1205821

119879minus2

2)

2

1205831015840

119879minus2(Σ1

119879minus2)minus1

(120583119879minus2


119868) = (1205821

119879minus2

2)

2

120572119879minus2

(B9)


119879minus2(119909119879minus2

119897119879minus2


119879minus2is

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1 119868

119886119879minus2

=1

119886119879minus2

(B10)



119902119879minus2


119909119879minus2



is

[Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] + 119886

119879minus21205742

119879minus2

minus [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]

= 1205732119879minus2

(B11)


is

minus 1205821

119879minus21205831015840

119879minus2(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)]

= 1205731119879minus2

(B12)


minus1205822

119879minus1

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(B13)

Thus

119881lowast

119879minus2(119909119879minus2

119897119879minus2

) =1

119886119879minus2

1199092


119879minus2119902119879minus2

119909119879minus2


119909119879minus2

119897119879minus2

+ 1205731119879minus2

119897119879minus2

+ 1205732119879minus2

1198972

119879minus2minus

1205822

119879minus1

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(B14)


1198681015840

119886119905minus1

(Σ1

119905minus1)minus1

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

=119889119905minus1

119886119905minus1

+ 120574119905

1198681015840(Σ1

119905minus1)minus1

119864 (119876119905minus1

119877119905minus1

)

119886119905minus1


= 120574119905minus1

+ 120574119905

1198681015840(Σ1

119905minus1)minus1

119864 (119876119905minus1

119877119905minus1

)

119886119905minus1


= 120574119905minus1


= 0

1198681015840

119886119905minus1

(Σ1

119905minus1)minus1

[120583119905minus1


119868] =119887119905minus1

119886119905minus1


= 0

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

119868

= 119886119905minus1

120574119905minus1

+ 120574119905

119864 (119876119905minus1

1198771015840

119905minus1) (Σ1

119905minus1)minus1

119868

119886119905minus1

= 119886119905minus1

120574119905minus1

(C1)


119905minus1

1015840Σ1




and 119897119905minus1

119909119905minus1


is

1198681015840

119886119905minus1

(Σ1

119905minus1)minus1 119868

119886119905minus1

=1

119886119905minus1

(C2)


is

1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

= 1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

minus 1205821

119905minus1120574119905minus1

1198681015840(Σ1

119905minus1)minus1

[120583119905minus1


119868]


= 1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

(C3)


is

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]1015840

sdot (Σ1

119905minus1)minus1

sdot [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

= [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] minus 119886119905minus1

1205742

119905minus1

(C4)


(1205821

119905minus1)2

4[120583119905minus1


119868]1015840

(Σ1

119905minus1)minus1

[120583119905minus1


119868] =(1205821

119905minus1)2

4120572119905minus1

(C5)




119905minus1is


is

1205821

119905minus1

2

1205831015840

119905minus1(Σ1

119905minus1)minus1

119868

119886119905minus1

=1205821

119905minus1119902119905minus1

2 (C6)


119909119905minus1

is 120574119905minus1


is

1205821

119905minus1

2[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1



119868) +1205821

119905minus1

21205831015840

119905minus1(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

(C7)


is

[Σ0

119905minus1+ 120574119905119864(119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

= [Σ0

119905minus1+ 120574119905119864(119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] minus 119886119905minus1

1205742

119905minus1

(C8)


(1205821

119905minus1

2)

2

120583119905minus1

(Σ1

119905minus1)minus1

(120583119905minus1


119868) = (1205821

119905minus1

2)

2

120572119905minus1

(C9)


119905minus1(119909119905minus1

119897119905minus1


119905minus1is 1119886



1

119905minus1119902119905minus1


119905minus1119909119905minus1



is

[Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] + 119886

119905minus11205742

119905minus1

minus [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]

= 1205732119905minus1

(C10)


is

minus 1205821

119905minus11205831015840

119905minus1(Σ1

119905minus1)minus1

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

+ (120582119905minus1

+ 1205731119905) 119864 (119876

119905minus1) = 1205731119905minus1

(C11)


(1205821

119894)2120572119894

Thus

119881lowast

119905minus1(119909119905minus1

119897119905minus1

) =1

119886119905minus1

1199092


119905minus1119902119905minus1

119909119905minus1


119909119905minus1

119897119905minus1

+ 1205731119905minus1

119897119905minus1

+ 1205732119905minus1

1198972

119905minus1minus

1

4

119879minus1

sum119894=119905minus1

(1205821

119894)2

120572119894

(C12)

D Proof of (62)


119879minus1into 119906

lowast

119879minus1

1015840Σ119879minus1

119906lowast


119906lowast

119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1= (1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840

119879minus1)


)minus1

(120593119879minus1

119897119879minus1

+120582119879minus1

2120598119879minus1

)

= 1205931015840


)minus1

120593119879minus1

1198972

119879minus1+ 120582119879minus1

1205981015840

119879minus1


)minus1

120593119879minus1

119897119879minus1

+(120582119879minus1

)2

4119872119879minus1

2 (1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840



119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1

(D1)

Thus

119881lowast

119879minus1(119909119879minus1

119897119879minus1


119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1+ Var (119876

119879minus1) 1198972

119879minus1


119904119879minus1

119909119879minus1

+ 120582119879minus1

119864 (119876119879minus1

) 119897119879minus1

= [Var (119876119879minus1

) minus 1205931015840


)minus1

120593119879minus1

] 1198972

119879minus1

+ 120582119879minus1

[119864 (119876119879minus1

) minus 1205981015840


)minus1

120593119879minus1

]



119904119879minus1

119909119879minus1


)2

4119872119879minus1

= 1205732119879minus1

1198972

119879minus1+ 1205731119879minus1

119897119879minus1


119904119879minus1

119909119879minus1


)2

4119872119879minus1

(D2)



lowast


119906lowast

119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2= (1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840

119879minus2)


)minus1

(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= 1205931015840


)minus1

120593119879minus2

1198972

119879minus2+ 1205821

119879minus21205981015840

119879minus2


)minus1

120593119879minus2

119897119879minus2

+(1205821

119879minus2)2

4119872119879minus2

2 1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840



119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2

(E1)Thus

119881lowast

119879minus2(119909119879minus2

119897119879minus2


119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2

+ [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

= 1205732119879minus2

1198972

119879minus2+ 1205731119879minus2

119897119879minus2

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

minus(1205821

119879minus2)2

4119872119879minus2

(E2)where1205731119879minus2

= (120582119879minus2

+ 1205731119879minus1

) 119864 (119876119879minus2

) minus 1205821

119879minus21205981015840


)minus1

120593119879minus2

1205732119879minus2

= Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2) minus 1205931015840


)minus1

120593119879minus2

(E3)


lowast

119905minus1that

119906lowast

119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1= (1205931015840

119879minus1119897119905minus1

+1205821

119905minus1

21205981015840

119879minus1)


)minus1

(120593119905minus1

119897119905minus1

+1205821

119905minus1

2120598119905minus1

)

= 1205931015840


)minus1

120593119905minus1

1198972

119905minus1+ 1205821

119905minus11205981015840

119879minus1


)minus1

120593119905minus1

119897119905minus1

+(1205821

119905minus1)2

4119872119905minus1

2 (1205931015840

119879minus1119897119905minus1

+1205821

119905minus1

21205981015840



119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1

(F1)

Thus

119881lowast

119905minus1(119909119905minus1

119897119905minus1


119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1

+ [Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] 1198972

119905minus1

+ [120582119905minus1

119864 (119876119905minus1

) + 1205731119905119864 (119876119905minus1

)] 119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

= 1205732119905minus1

1198972

119905minus1+ 1205731119905minus1

119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905minus1

(1205821

119894)2

4119872119894

(F2)

where

1205731119905minus1

= (120582119905minus1

+ 1205731119905) 119864 (119876

119905minus1) minus 1205821

119905minus11205981015840


)minus1

120593119905minus1

1205732119905minus1

= Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1) minus 1205931015840


)minus1

120593119905minus1

(F3)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












details)

119881lowast

119879minus2(119909119879minus2

119897119879minus2

) =1

119886119879minus2

1199092


119879minus2119902119879minus2

119909119879minus2


119909119879minus2

119897119879minus2

+ 1205731119879minus2

119897119879minus2

+ 1205732119879minus2

1198972

119879minus2


)2

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(34)


119881lowast

119905(119909119905 119897119905) =

1

119886119905

1199092

119905minus 1205821

119905119902119905119909119905minus 2120574119905119909119905119897119905

+ 1205731119905119897119905+ 12057321199051198972

119905minus

119879minus1

sum119894=119905

(1205821

119894)2

4120572119894

(35)


119905minus1and



min Var119905minus1

(119878119905) minus 120582119905minus1

119864119905minus1

(119878119905)

+ 119864119905minus1

(119881lowast

119905(119909119905 119897119905))

st 1198771015840

119905minus1119906119905minus1

= 119909119905

1198681015840119906119905minus1

= 119909119905minus1

119897119905= 119876119905minus1

119897119905minus1

119878119905= 119909119905minus 119897119905

(36)


119864119905minus1

(119881lowast

119905(119909119905 119897119905)) =

1

119886119905

119864119905minus1

(1199092

119905) minus 1205821

119905119902119905119864119905minus1

(119909119905)

minus 2120574119905119864119905minus1

(119909119905119897119905) + 1205731119905119864119905minus1

(119897119905)

+ 1205732119905119864119905minus1

(1198972

119905) minus

119879minus1

sum119894=119905

(1205821

119894)2

4120572119894

(37)


(119881lowast



min 119881119905minus1

(119909119905minus1

119897119905minus1

) = 1199061015840

119879minus1Σ1

119905minus1119906119905minus1

minus 2Φ1015840

119905minus1119906119905minus1

+ Ψ119905minus1

st 1198681015840119906119905minus1

= 119909119905minus1

(38)

where

Φ119905minus1

= [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] 119897119905minus1

+1205821

119905minus1

2120583119905minus1

Ψ119905minus1

= [Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] 1198972

119905minus1

+ [120582119905minus1

+ 1205731119905] 119864 (119876

119905minus1) 119897119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4120572119894

(39)



119871 (119906119905minus1

) = 1199061015840

119879minus1Σ1

119905minus1119906119905minus1

minus 2Φ1015840

119905minus1119906119905minus1

+ Ψ119905minus1

minus 120596119905(119909119905minus1

minus 1198681015840119906119905minus1

)

(40)


2Σ1

119905minus1119906119905minus1


+ 120596119905119868 = 0

119909119905minus1

minus 1198681015840119906119905minus1

= 0

(41)

Thus we have

119906lowast

119905minus1= (Σ1

119905minus1)minus1

Φ119905minus1

minus1

2120596119905(Σ1

119905minus1)minus1

119868 (42)

1198681015840119906lowast

119905minus1= 1198681015840(Σ1

119905minus1)minus1

Φ119905minus1

minus1

21205961199051198681015840(Σ1

119905minus1)minus1

119868

= 1198681015840(Σ1

119905minus1)minus1

Φ119905minus1

minus1

2120596119905119886119905minus1

= 119909119905minus1

(43)

which implies

120596119905= 2

1198681015840(Σ1

119905minus1)minus1

Φ119905minus1

119886119905minus1

minus 21

119886119905minus1

119909119905minus1

(44)


119906lowast

119905minus1=

(Σ1

119905minus1)minus1

119868

119886119905minus1

119909119905minus1

minus1198681015840(Σ1

119905minus1)minus1

Φ119905minus1

119886119905minus1

(Σ1

119905minus1)minus1

119868

+ (Σ1

119905minus1)minus1

Φ119905minus1

=(Σ1

119905minus1)minus1

119868

119886119905minus1

119909119905minus1

+1205821

119905minus1

2(Σ1

119905minus1)minus1

(120583119905minus1


119868)

+ (Σ1

119905minus1)minus1

[Σ0


119868 + 120574119905119864 (119876119905minus1

119877119905minus1

)] 119897119905minus1

(45)




119881lowast

119905minus1(119909119905minus1

119897119905minus1

) =1

119886119905minus1

1199092


119905minus1119902119905minus1

119909119905minus1


119909119905minus1

119897119905minus1

+ 1205731119905minus1

119897119905minus1

+ 1205732119905minus1

1198972

119905minus1minus

119879minus1

sum119894=119905minus1

(1205821

119894)2

4120572119894





119906lowast

119905=

(Σ1

119905)minus1

119868

119886119905

119909119905+

1205821

119905

2(Σ1

119905)minus1

(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1

(47)





119906lowast

119905=

(Σ1

119905)minus1

119868

119886119905

119909119905+

1205821

119905

2(Σ1

119905)minus1

(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1

(48)


119905= 1198681015840(Σ1

119905)minus1Σ0

119905=

0 120574119905= 119889119905119886119905= 0 and 120574


119906lowast

119905=

(Σ1

119905)minus1

119868

119886119905

119909119905+

1205821

119905

2(Σ1

119905)minus1

(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1

(49)




(Σ1

119905)minus1

(Σ0

119905minus 120574119905119868 + 120574119905+1

119864 (119876119905119877119905)) 119897119905 (50)



119905


min119906119905

Var119905(119878119905+1

) minus 120582119905119864119905(119878119905+1

)

st 1198771015840

119905119906119905= 119909119905+1

1198681015840119906119905= 119909119905

119897119905+1

= 119876119905119897119905

119878119905+1

= 119909119905+1

minus 119897119905+1

(51)


119906119898119910

119905=

(Σ119905)minus1

119868

119886119905

119909119905+

120582119905

2(Σ119905)minus1

(120583119905minus 119902119905119868)

+ (Σ119905)minus1

(Σ0

119905minus 120574119905119868) 119897119905 119905 = 0 119879 minus 1

(52)


119905119886119905


119905rarr 0


119906lowast

119905=

(Σ1

119905)minus1

119868

119886119905

119909119905+ (Σ1

119905)minus1

(Σ0

119905minus 120574119905119868 + 120574119905+1

119864 (119876119905119877119905)) 119897119905

119905 = 0 119879 minus 1

119906119898119910

119905=

(Σ119905)minus1

119868

119886119905

119909119905+ (Σ119905)minus1

(Σ0

119905minus 120574119905119868) 119897119905 119905 = 0 119879 minus 1

(53)







119909119905= 1198751015840

119905minus1119906119905minus1

+ 119904119905minus1

119909119905minus1

(54)



0(1198781) + 1198640(1205881(1198782) + 1198641(1205882(1198783) + sdot sdot sdot

+119864119879minus3

(120588119879minus2

(119878119879minus1

) + 119864119879minus2

(120588119879minus1

(119878119879))) ))

st 119909119905+1

= 1198751015840

119905119906119905+ 119904119905119909119905

119897119905+1

= 119876119905119897119905

119878119905+1

= 119909119905+1

minus 119897119905+1

119905 = 0 1 119879 minus 1

(55)



119906lowast

119905= Σminus1

119905120593119905119897119905+

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (56)

where 120593119905= 119864(119875

119905119876119905) minus 119864(119875

119905)119864(119876119905) and 120598

119905= 119864(119875

119905)




119879minus1= 120582119879minus1

1205821

119905= 120582119905+ 1205821

119905+1119904119905+1

1205731119879minus1

= 120582119879minus1

[119864 (119876119879minus1

) minus 1205981015840


)minus1

120593119879minus1

] 1205731119905

= (120582119905+ 1205731119905+1

) 119864(119876119905) minus 1205821

1199051205981015840

119905(Σ119905)minus1120593119905

1205732119879minus1

= Var(119876119879minus1

) minus 1205931015840


)minus1

120593119879minus1

1205732119905

= Var(119876119905) + 1205732119905+1

119864(1198762

119905) minus 1205931015840

119905(Σ119905)minus1120593119905




min119906119879minus1

Var119879minus1

(119878119879) minus 120582119879minus1

119864119879minus1

(119878119879)

st 119909119879

= 119904119879minus1

119909119879minus1

+ 1198751015840

119879minus1119906119879minus1

119897119879

= 119876119879minus1

119897119879minus1

119878119879= 119909119879minus 119897119879

(57)


min119906119879minus1

119881119879minus1

(119909119879minus1

119897119879minus1

) (58)

where

119881119879minus1

(119909119879minus1

119897119879minus1

) = 1199061015840


119906119879minus1

minus 2(1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840

119879minus1)119906119879minus1

+ Var (119876119879minus1

) 1198972


119904119879minus1

119909119879minus1

+ 120582119879minus1

119864 (119876119879minus1

) 119897119879minus1

(59)


119889 (119881119879minus1

(119909119879minus1

119897119879minus1

))

119889 (119906119879minus1

)= 2Σ119879minus1

119906119879minus1

minus 2(120593119879minus1

119897119879minus1

+120582119879minus1

2120598119879minus1

)

= 0

(60)

which implies

119906lowast

119879minus1= (Σ119879minus1

)minus1

[120593119879minus1

119897119879minus1

+120582119879minus1

2120598119879minus1

] (61)




119881lowast

119879minus1(119909119879minus1

119897119879minus1

) = 1205732119879minus1

1198972

119879minus1+ 1205731119879minus1

119897119879minus1


119904119879minus1

119909119879minus1


)2

4119872119879minus1

(62)


= 1205981015840


119879minus1120598119879minus1




min Var119879minus2

(119878119879minus1

) minus 120582119879minus2

119864119879minus2

(119878119879minus1

)

+ 119864119879minus2

(119881lowast

119879minus1(119909119879minus1

119897119879minus1

))

st 119909119879minus1

= 119904119879minus2

119909119879minus2

+ 1198751015840

119879minus2119906119879minus2

119897119879minus1

= 119876119879minus2

119897119879minus2

119878119879minus1

= 119909119879minus1


(63)


119864119879minus2

(119881lowast

119879minus1(119909119879minus1

119897119879minus1

)) = 1205732119879minus1

119864 (1198762

119879minus2) 1198972

119879minus2

+ 1205731119879minus1

119864 (119876119879minus2

) 119897119879minus2


119904119879minus1

119904119879minus2

119909119879minus2


119904119879minus1

1205981015840

119879minus2119906119879minus2


)2

4119872119879minus1

(64)


(119881lowast

119879minus1(119909119879minus1

119897119879minus1


min119906119879minus2

119881119879minus2

(119909119879minus2

119897119879minus2

) (65)


(119909119879minus2

119897119879minus2

) = 1199061015840


119906119879minus2

minus 2(1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840

119879minus2)119906119879minus2

+ [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

(66)


119889 (119881119879minus2

(119909119879minus2

119897119879minus2

))

119889 (119906119879minus2

)= 2Σ119879minus2

119906119879minus2

minus 2(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= 0

(67)


Thus

119906lowast

119879minus2= (Σ119879minus2

)minus1

(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= (Σ119879minus2

)minus1

120593119879minus2

119897119879minus2

+1205821

119879minus2

2(Σ119879minus2

)minus1

120598119879minus2

(68)




119881lowast

119879minus2(119909119879minus2

119897119879minus2

) = 1205732119879minus2

1198972

119879minus2+ 1205731119879minus2

119897119879minus2

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

minus(1205821

119879minus2)2

4119872119879minus2

(69)


= 1205981015840


119879minus2120598119879minus2


both (56) and

119881lowast

119905(119909119905 119897119905) = 12057321199051198972

119905+ 1205731119905119897119905minus 1205821

119905119904119905119909119905minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

(70)

hold where 119872119894= 1205981015840

119894Σminus1



119905minus1and



min Var119905minus1

(119878119905) minus 120582119905minus1

119864119905minus1

(119878119905) + 119864119905minus1

(119881lowast

119905(119909119905 119897119905))

st 119909119905= 119904119905minus1

119909119905minus1

+ 1198751015840

119905minus1119906119905minus1

119897119905= 119876119905minus1

119897119905minus1

119878119905= 119909119905minus 119897119905

(71)


119864119905minus1

(119881lowast

119905(119909119905 119897119905)) = 120573

2119905119864 (1198762

119905minus1) 1198972

119905minus1+ 1205731119905119864 (119876119905minus1

) 119897119905

minus 1205821

119905119904119905119904119905minus1

119909119905minus1

minus 1205821

1199051199041199051205981015840

119905minus1119906119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

(72)


(119881lowast



119881119905minus1

(119909119905minus1

119897119905minus1

) (73)

where

119881119905minus1

(119909119905minus1

119897119905minus1

) = 1199061015840


119906119905minus1

minus 2(1205931015840

119905minus1119897119905minus1

+1205821

119905minus1

21205981015840

119905minus1)119906119905minus1

+ [Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] 1198972

119905minus1

+ [120582119905minus1

119864 (119876119905minus1

) + 1205731119905119864 (119876119905minus1

)] 119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

(74)


119889 (119881119905minus1

(119909119905minus1

119897119905minus1

))

119889 (119906119905minus1

)= 2Σ119905minus1

119906119905minus1

minus 2(120593119905minus1

119897119905minus1

+1205821

119905minus1

2120598119905minus1

)

= 0

(75)

which implies

119906lowast


119905minus1120593119905minus1

119897119905minus1

+1205821

119905minus1

2Σminus1

119905minus1120598119905minus1

(76)




119881lowast

119905minus1(119909119905minus1

119897119905minus1

) = 1205732119905minus1

1198972

119905minus1+ 1205731119905minus1

119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905minus1

(1205821

119894)2

4119872119894

(77)







119906lowast

119905=

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (78)






119906lowast

119905=

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (79)


120593119905= 119864 (119875

119905119876119905) minus 119864 (119875

119905) 119864 (119876

119905)

= 119864 ((119877119905minus 119904119905119868)119876119905) minus 119864 (119877

119905minus 119904119905119868) 119864 (119876

119905)

= 119864 (119877119905119876119905) minus 119864 (119877

119905) 119864 (119876

119905)

= Σ0

119905

= 0

(80)



119906lowast

119905=

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (81)




Σminus1

119905120593119905119897119905 (82)



min119906119905

Var119905(119878119905+1

) minus 120582119905119864119905(119878119905+1

)

st 119909119905+1

= 119904119905119909119905+ 1198751015840

119905119906119905

119897119905+1

= 119876119905119897119905

119878119905+1

= 119909119905+1

minus 119897119905+1

(83)


119906119898119910

119905= Σminus1

119905120593119905119897119905+

120582119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (84)




119906lowast

119905= Σminus1

119905120593119905119897119905 119905 = 0 119879 minus 1 (85)






119905


lowast

119905

0 (

11941198970+ 0903119909

0+ 3736

01071198970minus 0066119909

0+ 0203

minus05771198970+ 0163119909

0minus 3939

) (

09031199090+ 3736

minus00661199090+ 0203

01631199090minus 3939

)

1 (

minus03721198971+ 1018119909

1+ 1743

01981198971minus 0060119909

1+ 0095

minus06331198971+ 0042119909

1minus 1838

) (

10181199091+ 1743

minus00601199091+ 0095

00421199091minus 1838

)




my119905

0 (

09031199090+ 1194119897

0+ 3736

minus00661199090+ 0107119897

0+ 0203

01631199090minus 0577119897

0minus 3939

) (

10181199090+ 0449119897

0+ 1743

minus00601199090+ 0150119897

0+ 0095

00421199090minus 0599119897

0minus 1838

)

1 (

10181199091minus 0372119897

1+ 1743

minus00601199091+ 0198119897

1+ 0095

00421199091minus 0633119897

1minus 1838

) (

10181199091+ 0449119897

1+ 1743

minus00601199091+ 0150119897

1+ 0095

00421199091minus 0599119897

1minus 1838

)



119905=

(1162 1246 1228) and

Σ119905= (

00146 00187 00145

00187 00854 00104

00145 00104 00289

) (86)






119905=

05 and 120582119905







my119905

0 (

09031199090+ 1194119897

0

minus00661199090+ 0107119897

0

01631199090minus 0577119897

0

) (

10181199090+ 0449119897

0

minus00601199090+ 0150119897

0

00421199090minus 0599119897

0

)

1 (

10181199091minus 0372119897

1

minus00601199091+ 0198119897

1

00421199091minus 0633119897

1

) (

10181199091+ 0449119897

1

minus00601199091+ 0150119897

1

00421199091minus 0599119897

1

)




my119905

0 (

04441198970+ 7889

01501198970minus 0062

minus05991198970minus 3572

) (

04441198970+ 3867

01501198970minus 0031

minus05991198970minus 1751

)

1 (

04441198971+ 3867

01501198971minus 0031

minus05991198971minus 1751

) (

04441198971+ 3867

01501198971minus 0031

minus05991198971minus 1751

)




my119905

0 (

04441198970

01501198970

minus05991198970

) (

04441198970

01501198970

minus05991198970

)

1 (

04441198971

01501198971

minus05991198971

) (

04441198971

01501198971

minus05991198971

)


119905= 120583119905minus 119904119905119868 =

(0122 0206 0188)1015840 and120593

119905= Σ0

119905= (00006 00149 00050)





6 Conclusion


Appendices

A Proof of (20)


119879minus1into 119906

lowast

119879minus1

1015840Σ1


119879minus1gives

119906lowast

119879minus1

1015840

Σ1


119879minus1=

119868

119886119879minus1

119909119879minus1

+ [Σ0


119868] 119897119879minus1

+120582119879minus1

2[120583119879minus1


119868]

1015840

(Σ1

119879minus1)minus1

times 119868

119886119879minus1

119909119879minus1

+ [Σ0


119868]


+120582119879minus1

2[120583119879minus1


119868]

=1

119886119879minus1

1199092

119879minus1+ [Σ0


119868]1015840

times (Σ1

119879minus1)minus1

[Σ0


119868] 1198972

119879minus1

+ 120582119879minus1

[Σ0


119868]1015840

times (Σ1

119879minus1)minus1

[120583119879minus1


119868] 119897119879minus1

+(120582119879minus1

)2

4[120583119879minus1


119868]1015840

times (Σ1

119879minus1)minus1

[120583119879minus1


119868]

=1

119886119879minus1

1199092

119879minus1+ [(Σ

0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0

119879minus1


1205742

119879minus1] 1198972

119879minus1

+ [(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1


120574119879minus1

]


119897119879minus1

+(120582119879minus1

)2

4120572119879minus1

(A1)



0

119879minus1)1015840

119897119879minus1

+

(120582119879minus1

2)1205831015840



2 [(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840


119879minus1

=2

119886119879minus1

[(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1

119868119909119879minus1

+ 2 [(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1


times [Σ0


119868] 119897119879minus1

+ 120582119879minus1

[(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1



119868]

= 2120574119879minus1

119897119879minus1

119909119879minus1

+ 119902119879minus1

120582119879minus1

119909119879minus1

+ 2 [(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0


119889119879minus1

] 1198972

119879minus1

+ 2120582119879minus1

[(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1


120574119879minus1

] 119897119879minus1

+(120582119879minus1

)2

2120572119879minus1

(A2)

Thus119881lowast

119879minus1(119909119879minus1

119897119879minus1

)

=1

119886119879minus1

1199092


119902119879minus1

119909119879minus1


119909119879minus1

119897119879minus1

+ 120582119879minus1

[119864 (119876119879minus1

) minus (Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1

+119887119879minus1

120574119879minus1

] 119897119879minus1

+ [Var (119876119879minus1

) minus (Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0

119879minus1

+120574119879minus1

119889119879minus1

] 1198972

119879minus1minus

(120582119879minus1

)2

4120572119879minus1

=1

119886119879minus1

1199092


119902119879minus1

119909119879minus1


119909119879minus1

119897119879minus1

+ 1205731119879minus1

119897119879minus1

+ 1205732119879minus1

1198972

119879minus1minus

(120582119879minus1

)2

4120572119879minus1

(A3)


1198681015840

119886119879minus2

(Σ1

119879minus2)minus1

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

=119889119879minus2

119886119879minus2

+ 120574119879minus1

1198681015840(Σ1

119879minus2)minus1

119864 (119876119879minus2

119877119879minus2

)

119886119879minus2


= 120574119879minus2

+ 120574119879minus1

1198681015840(Σ1

119879minus2)minus1

119864 (119876119879minus2

119877119879minus2

)

119886119879minus2


= 120574119879minus2


= 0

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1

[120583119879minus2


119868] =119887119879minus2

119886119879minus2


= 0

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

119868

= 119886119879minus2

120574119879minus2

+ 120574119879minus1

119864 (119876119879minus2

1198771015840

119879minus2) (Σ1

119879minus2)minus1

119868

119886119879minus2

= 119886119879minus2

120574119879minus2

(B1)


119879minus2

1015840Σ1




and 119897119879minus2

119909119879minus2

in 119906lowast

119879minus2

1015840Σ1



119879minus2is

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1 119868

119886119879minus2

=1

119886119879minus2

(B2)


is

1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

= 1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

minus 1205821

119879minus2120574119879minus2

1198681015840(Σ1

119879minus2)minus1

[120583119879minus2


119868]

= 1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

(B3)


is

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]1015840

sdot (Σ1

119879minus2)minus1

sdot [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

= [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]


1205742

119879minus2

(B4)


(1205821

119879minus2)2

4[120583119879minus2


119868]1015840

(Σ1

119879minus2)minus1

[120583119879minus2


119868]

=(1205821

119879minus2)2

4120572119879minus2

(B5)




119879minus2


is

1205821

119879minus2

2

1205831015840

119879minus2(Σ1

119879minus2)minus1

119868

119886119879minus2

=1205821

119879minus2119902119879minus2

2 (B6)



119909119879minus2

is 120574119879minus2


is

1205821

119879minus2

2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

(120583119879minus2


119868)

+1205821

119879minus2

21205831015840

119879minus2(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

(B7)


is

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

= [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)] minus 119886119879minus2

1205742

119879minus2

(B8)


(1205821

119879minus2

2)

2

1205831015840

119879minus2(Σ1

119879minus2)minus1

(120583119879minus2


119868) = (1205821

119879minus2

2)

2

120572119879minus2

(B9)


119879minus2(119909119879minus2

119897119879minus2


119879minus2is

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1 119868

119886119879minus2

=1

119886119879minus2

(B10)



119902119879minus2


119909119879minus2



is

[Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] + 119886

119879minus21205742

119879minus2

minus [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]

= 1205732119879minus2

(B11)


is

minus 1205821

119879minus21205831015840

119879minus2(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)]

= 1205731119879minus2

(B12)


minus1205822

119879minus1

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(B13)

Thus

119881lowast

119879minus2(119909119879minus2

119897119879minus2

) =1

119886119879minus2

1199092


119879minus2119902119879minus2

119909119879minus2


119909119879minus2

119897119879minus2

+ 1205731119879minus2

119897119879minus2

+ 1205732119879minus2

1198972

119879minus2minus

1205822

119879minus1

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(B14)


1198681015840

119886119905minus1

(Σ1

119905minus1)minus1

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

=119889119905minus1

119886119905minus1

+ 120574119905

1198681015840(Σ1

119905minus1)minus1

119864 (119876119905minus1

119877119905minus1

)

119886119905minus1


= 120574119905minus1

+ 120574119905

1198681015840(Σ1

119905minus1)minus1

119864 (119876119905minus1

119877119905minus1

)

119886119905minus1


= 120574119905minus1


= 0

1198681015840

119886119905minus1

(Σ1

119905minus1)minus1

[120583119905minus1


119868] =119887119905minus1

119886119905minus1


= 0

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

119868

= 119886119905minus1

120574119905minus1

+ 120574119905

119864 (119876119905minus1

1198771015840

119905minus1) (Σ1

119905minus1)minus1

119868

119886119905minus1

= 119886119905minus1

120574119905minus1

(C1)


119905minus1

1015840Σ1




and 119897119905minus1

119909119905minus1


is

1198681015840

119886119905minus1

(Σ1

119905minus1)minus1 119868

119886119905minus1

=1

119886119905minus1

(C2)


is

1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

= 1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

minus 1205821

119905minus1120574119905minus1

1198681015840(Σ1

119905minus1)minus1

[120583119905minus1


119868]


= 1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

(C3)


is

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]1015840

sdot (Σ1

119905minus1)minus1

sdot [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

= [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] minus 119886119905minus1

1205742

119905minus1

(C4)


(1205821

119905minus1)2

4[120583119905minus1


119868]1015840

(Σ1

119905minus1)minus1

[120583119905minus1


119868] =(1205821

119905minus1)2

4120572119905minus1

(C5)




119905minus1is


is

1205821

119905minus1

2

1205831015840

119905minus1(Σ1

119905minus1)minus1

119868

119886119905minus1

=1205821

119905minus1119902119905minus1

2 (C6)


119909119905minus1

is 120574119905minus1


is

1205821

119905minus1

2[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1



119868) +1205821

119905minus1

21205831015840

119905minus1(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

(C7)


is

[Σ0

119905minus1+ 120574119905119864(119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

= [Σ0

119905minus1+ 120574119905119864(119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] minus 119886119905minus1

1205742

119905minus1

(C8)


(1205821

119905minus1

2)

2

120583119905minus1

(Σ1

119905minus1)minus1

(120583119905minus1


119868) = (1205821

119905minus1

2)

2

120572119905minus1

(C9)


119905minus1(119909119905minus1

119897119905minus1


119905minus1is 1119886



1

119905minus1119902119905minus1


119905minus1119909119905minus1



is

[Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] + 119886

119905minus11205742

119905minus1

minus [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]

= 1205732119905minus1

(C10)


is

minus 1205821

119905minus11205831015840

119905minus1(Σ1

119905minus1)minus1

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

+ (120582119905minus1

+ 1205731119905) 119864 (119876

119905minus1) = 1205731119905minus1

(C11)


(1205821

119894)2120572119894

Thus

119881lowast

119905minus1(119909119905minus1

119897119905minus1

) =1

119886119905minus1

1199092


119905minus1119902119905minus1

119909119905minus1


119909119905minus1

119897119905minus1

+ 1205731119905minus1

119897119905minus1

+ 1205732119905minus1

1198972

119905minus1minus

1

4

119879minus1

sum119894=119905minus1

(1205821

119894)2

120572119894

(C12)

D Proof of (62)


119879minus1into 119906

lowast

119879minus1

1015840Σ119879minus1

119906lowast


119906lowast

119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1= (1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840

119879minus1)


)minus1

(120593119879minus1

119897119879minus1

+120582119879minus1

2120598119879minus1

)

= 1205931015840


)minus1

120593119879minus1

1198972

119879minus1+ 120582119879minus1

1205981015840

119879minus1


)minus1

120593119879minus1

119897119879minus1

+(120582119879minus1

)2

4119872119879minus1

2 (1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840



119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1

(D1)

Thus

119881lowast

119879minus1(119909119879minus1

119897119879minus1


119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1+ Var (119876

119879minus1) 1198972

119879minus1


119904119879minus1

119909119879minus1

+ 120582119879minus1

119864 (119876119879minus1

) 119897119879minus1

= [Var (119876119879minus1

) minus 1205931015840


)minus1

120593119879minus1

] 1198972

119879minus1

+ 120582119879minus1

[119864 (119876119879minus1

) minus 1205981015840


)minus1

120593119879minus1

]



119904119879minus1

119909119879minus1


)2

4119872119879minus1

= 1205732119879minus1

1198972

119879minus1+ 1205731119879minus1

119897119879minus1


119904119879minus1

119909119879minus1


)2

4119872119879minus1

(D2)



lowast


119906lowast

119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2= (1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840

119879minus2)


)minus1

(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= 1205931015840


)minus1

120593119879minus2

1198972

119879minus2+ 1205821

119879minus21205981015840

119879minus2


)minus1

120593119879minus2

119897119879minus2

+(1205821

119879minus2)2

4119872119879minus2

2 1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840



119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2

(E1)Thus

119881lowast

119879minus2(119909119879minus2

119897119879minus2


119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2

+ [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

= 1205732119879minus2

1198972

119879minus2+ 1205731119879minus2

119897119879minus2

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

minus(1205821

119879minus2)2

4119872119879minus2

(E2)where1205731119879minus2

= (120582119879minus2

+ 1205731119879minus1

) 119864 (119876119879minus2

) minus 1205821

119879minus21205981015840


)minus1

120593119879minus2

1205732119879minus2

= Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2) minus 1205931015840


)minus1

120593119879minus2

(E3)


lowast

119905minus1that

119906lowast

119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1= (1205931015840

119879minus1119897119905minus1

+1205821

119905minus1

21205981015840

119879minus1)


)minus1

(120593119905minus1

119897119905minus1

+1205821

119905minus1

2120598119905minus1

)

= 1205931015840


)minus1

120593119905minus1

1198972

119905minus1+ 1205821

119905minus11205981015840

119879minus1


)minus1

120593119905minus1

119897119905minus1

+(1205821

119905minus1)2

4119872119905minus1

2 (1205931015840

119879minus1119897119905minus1

+1205821

119905minus1

21205981015840



119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1

(F1)

Thus

119881lowast

119905minus1(119909119905minus1

119897119905minus1


119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1

+ [Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] 1198972

119905minus1

+ [120582119905minus1

119864 (119876119905minus1

) + 1205731119905119864 (119876119905minus1

)] 119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

= 1205732119905minus1

1198972

119905minus1+ 1205731119905minus1

119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905minus1

(1205821

119894)2

4119872119894

(F2)

where

1205731119905minus1

= (120582119905minus1

+ 1205731119905) 119864 (119876

119905minus1) minus 1205821

119905minus11205981015840


)minus1

120593119905minus1

1205732119905minus1

= Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1) minus 1205931015840


)minus1

120593119905minus1

(F3)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119906lowast

119905=

(Σ1

119905)minus1

119868

119886119905

119909119905+

1205821

119905

2(Σ1

119905)minus1

(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1

(47)





119906lowast

119905=

(Σ1

119905)minus1

119868

119886119905

119909119905+

1205821

119905

2(Σ1

119905)minus1

(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1

(48)


119905= 1198681015840(Σ1

119905)minus1Σ0

119905=

0 120574119905= 119889119905119886119905= 0 and 120574


119906lowast

119905=

(Σ1

119905)minus1

119868

119886119905

119909119905+

1205821

119905

2(Σ1

119905)minus1

(120583119905minus 119902119905119868) 119905 = 0 119879 minus 1

(49)




(Σ1

119905)minus1

(Σ0

119905minus 120574119905119868 + 120574119905+1

119864 (119876119905119877119905)) 119897119905 (50)



119905


min119906119905

Var119905(119878119905+1

) minus 120582119905119864119905(119878119905+1

)

st 1198771015840

119905119906119905= 119909119905+1

1198681015840119906119905= 119909119905

119897119905+1

= 119876119905119897119905

119878119905+1

= 119909119905+1

minus 119897119905+1

(51)


119906119898119910

119905=

(Σ119905)minus1

119868

119886119905

119909119905+

120582119905

2(Σ119905)minus1

(120583119905minus 119902119905119868)

+ (Σ119905)minus1

(Σ0

119905minus 120574119905119868) 119897119905 119905 = 0 119879 minus 1

(52)


119905119886119905


119905rarr 0


119906lowast

119905=

(Σ1

119905)minus1

119868

119886119905

119909119905+ (Σ1

119905)minus1

(Σ0

119905minus 120574119905119868 + 120574119905+1

119864 (119876119905119877119905)) 119897119905

119905 = 0 119879 minus 1

119906119898119910

119905=

(Σ119905)minus1

119868

119886119905

119909119905+ (Σ119905)minus1

(Σ0

119905minus 120574119905119868) 119897119905 119905 = 0 119879 minus 1

(53)







119909119905= 1198751015840

119905minus1119906119905minus1

+ 119904119905minus1

119909119905minus1

(54)



0(1198781) + 1198640(1205881(1198782) + 1198641(1205882(1198783) + sdot sdot sdot

+119864119879minus3

(120588119879minus2

(119878119879minus1

) + 119864119879minus2

(120588119879minus1

(119878119879))) ))

st 119909119905+1

= 1198751015840

119905119906119905+ 119904119905119909119905

119897119905+1

= 119876119905119897119905

119878119905+1

= 119909119905+1

minus 119897119905+1

119905 = 0 1 119879 minus 1

(55)



119906lowast

119905= Σminus1

119905120593119905119897119905+

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (56)

where 120593119905= 119864(119875

119905119876119905) minus 119864(119875

119905)119864(119876119905) and 120598

119905= 119864(119875

119905)




119879minus1= 120582119879minus1

1205821

119905= 120582119905+ 1205821

119905+1119904119905+1

1205731119879minus1

= 120582119879minus1

[119864 (119876119879minus1

) minus 1205981015840


)minus1

120593119879minus1

] 1205731119905

= (120582119905+ 1205731119905+1

) 119864(119876119905) minus 1205821

1199051205981015840

119905(Σ119905)minus1120593119905

1205732119879minus1

= Var(119876119879minus1

) minus 1205931015840


)minus1

120593119879minus1

1205732119905

= Var(119876119905) + 1205732119905+1

119864(1198762

119905) minus 1205931015840

119905(Σ119905)minus1120593119905




min119906119879minus1

Var119879minus1

(119878119879) minus 120582119879minus1

119864119879minus1

(119878119879)

st 119909119879

= 119904119879minus1

119909119879minus1

+ 1198751015840

119879minus1119906119879minus1

119897119879

= 119876119879minus1

119897119879minus1

119878119879= 119909119879minus 119897119879

(57)


min119906119879minus1

119881119879minus1

(119909119879minus1

119897119879minus1

) (58)

where

119881119879minus1

(119909119879minus1

119897119879minus1

) = 1199061015840


119906119879minus1

minus 2(1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840

119879minus1)119906119879minus1

+ Var (119876119879minus1

) 1198972


119904119879minus1

119909119879minus1

+ 120582119879minus1

119864 (119876119879minus1

) 119897119879minus1

(59)


119889 (119881119879minus1

(119909119879minus1

119897119879minus1

))

119889 (119906119879minus1

)= 2Σ119879minus1

119906119879minus1

minus 2(120593119879minus1

119897119879minus1

+120582119879minus1

2120598119879minus1

)

= 0

(60)

which implies

119906lowast

119879minus1= (Σ119879minus1

)minus1

[120593119879minus1

119897119879minus1

+120582119879minus1

2120598119879minus1

] (61)




119881lowast

119879minus1(119909119879minus1

119897119879minus1

) = 1205732119879minus1

1198972

119879minus1+ 1205731119879minus1

119897119879minus1


119904119879minus1

119909119879minus1


)2

4119872119879minus1

(62)


= 1205981015840


119879minus1120598119879minus1




min Var119879minus2

(119878119879minus1

) minus 120582119879minus2

119864119879minus2

(119878119879minus1

)

+ 119864119879minus2

(119881lowast

119879minus1(119909119879minus1

119897119879minus1

))

st 119909119879minus1

= 119904119879minus2

119909119879minus2

+ 1198751015840

119879minus2119906119879minus2

119897119879minus1

= 119876119879minus2

119897119879minus2

119878119879minus1

= 119909119879minus1


(63)


119864119879minus2

(119881lowast

119879minus1(119909119879minus1

119897119879minus1

)) = 1205732119879minus1

119864 (1198762

119879minus2) 1198972

119879minus2

+ 1205731119879minus1

119864 (119876119879minus2

) 119897119879minus2


119904119879minus1

119904119879minus2

119909119879minus2


119904119879minus1

1205981015840

119879minus2119906119879minus2


)2

4119872119879minus1

(64)


(119881lowast

119879minus1(119909119879minus1

119897119879minus1


min119906119879minus2

119881119879minus2

(119909119879minus2

119897119879minus2

) (65)


(119909119879minus2

119897119879minus2

) = 1199061015840


119906119879minus2

minus 2(1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840

119879minus2)119906119879minus2

+ [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

(66)


119889 (119881119879minus2

(119909119879minus2

119897119879minus2

))

119889 (119906119879minus2

)= 2Σ119879minus2

119906119879minus2

minus 2(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= 0

(67)


Thus

119906lowast

119879minus2= (Σ119879minus2

)minus1

(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= (Σ119879minus2

)minus1

120593119879minus2

119897119879minus2

+1205821

119879minus2

2(Σ119879minus2

)minus1

120598119879minus2

(68)




119881lowast

119879minus2(119909119879minus2

119897119879minus2

) = 1205732119879minus2

1198972

119879minus2+ 1205731119879minus2

119897119879minus2

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

minus(1205821

119879minus2)2

4119872119879minus2

(69)


= 1205981015840


119879minus2120598119879minus2


both (56) and

119881lowast

119905(119909119905 119897119905) = 12057321199051198972

119905+ 1205731119905119897119905minus 1205821

119905119904119905119909119905minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

(70)

hold where 119872119894= 1205981015840

119894Σminus1



119905minus1and



min Var119905minus1

(119878119905) minus 120582119905minus1

119864119905minus1

(119878119905) + 119864119905minus1

(119881lowast

119905(119909119905 119897119905))

st 119909119905= 119904119905minus1

119909119905minus1

+ 1198751015840

119905minus1119906119905minus1

119897119905= 119876119905minus1

119897119905minus1

119878119905= 119909119905minus 119897119905

(71)


119864119905minus1

(119881lowast

119905(119909119905 119897119905)) = 120573

2119905119864 (1198762

119905minus1) 1198972

119905minus1+ 1205731119905119864 (119876119905minus1

) 119897119905

minus 1205821

119905119904119905119904119905minus1

119909119905minus1

minus 1205821

1199051199041199051205981015840

119905minus1119906119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

(72)


(119881lowast



119881119905minus1

(119909119905minus1

119897119905minus1

) (73)

where

119881119905minus1

(119909119905minus1

119897119905minus1

) = 1199061015840


119906119905minus1

minus 2(1205931015840

119905minus1119897119905minus1

+1205821

119905minus1

21205981015840

119905minus1)119906119905minus1

+ [Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] 1198972

119905minus1

+ [120582119905minus1

119864 (119876119905minus1

) + 1205731119905119864 (119876119905minus1

)] 119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

(74)


119889 (119881119905minus1

(119909119905minus1

119897119905minus1

))

119889 (119906119905minus1

)= 2Σ119905minus1

119906119905minus1

minus 2(120593119905minus1

119897119905minus1

+1205821

119905minus1

2120598119905minus1

)

= 0

(75)

which implies

119906lowast


119905minus1120593119905minus1

119897119905minus1

+1205821

119905minus1

2Σminus1

119905minus1120598119905minus1

(76)




119881lowast

119905minus1(119909119905minus1

119897119905minus1

) = 1205732119905minus1

1198972

119905minus1+ 1205731119905minus1

119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905minus1

(1205821

119894)2

4119872119894

(77)







119906lowast

119905=

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (78)






119906lowast

119905=

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (79)


120593119905= 119864 (119875

119905119876119905) minus 119864 (119875

119905) 119864 (119876

119905)

= 119864 ((119877119905minus 119904119905119868)119876119905) minus 119864 (119877

119905minus 119904119905119868) 119864 (119876

119905)

= 119864 (119877119905119876119905) minus 119864 (119877

119905) 119864 (119876

119905)

= Σ0

119905

= 0

(80)



119906lowast

119905=

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (81)




Σminus1

119905120593119905119897119905 (82)



min119906119905

Var119905(119878119905+1

) minus 120582119905119864119905(119878119905+1

)

st 119909119905+1

= 119904119905119909119905+ 1198751015840

119905119906119905

119897119905+1

= 119876119905119897119905

119878119905+1

= 119909119905+1

minus 119897119905+1

(83)


119906119898119910

119905= Σminus1

119905120593119905119897119905+

120582119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (84)




119906lowast

119905= Σminus1

119905120593119905119897119905 119905 = 0 119879 minus 1 (85)






119905


lowast

119905

0 (

11941198970+ 0903119909

0+ 3736

01071198970minus 0066119909

0+ 0203

minus05771198970+ 0163119909

0minus 3939

) (

09031199090+ 3736

minus00661199090+ 0203

01631199090minus 3939

)

1 (

minus03721198971+ 1018119909

1+ 1743

01981198971minus 0060119909

1+ 0095

minus06331198971+ 0042119909

1minus 1838

) (

10181199091+ 1743

minus00601199091+ 0095

00421199091minus 1838

)




my119905

0 (

09031199090+ 1194119897

0+ 3736

minus00661199090+ 0107119897

0+ 0203

01631199090minus 0577119897

0minus 3939

) (

10181199090+ 0449119897

0+ 1743

minus00601199090+ 0150119897

0+ 0095

00421199090minus 0599119897

0minus 1838

)

1 (

10181199091minus 0372119897

1+ 1743

minus00601199091+ 0198119897

1+ 0095

00421199091minus 0633119897

1minus 1838

) (

10181199091+ 0449119897

1+ 1743

minus00601199091+ 0150119897

1+ 0095

00421199091minus 0599119897

1minus 1838

)



119905=

(1162 1246 1228) and

Σ119905= (

00146 00187 00145

00187 00854 00104

00145 00104 00289

) (86)






119905=

05 and 120582119905







my119905

0 (

09031199090+ 1194119897

0

minus00661199090+ 0107119897

0

01631199090minus 0577119897

0

) (

10181199090+ 0449119897

0

minus00601199090+ 0150119897

0

00421199090minus 0599119897

0

)

1 (

10181199091minus 0372119897

1

minus00601199091+ 0198119897

1

00421199091minus 0633119897

1

) (

10181199091+ 0449119897

1

minus00601199091+ 0150119897

1

00421199091minus 0599119897

1

)




my119905

0 (

04441198970+ 7889

01501198970minus 0062

minus05991198970minus 3572

) (

04441198970+ 3867

01501198970minus 0031

minus05991198970minus 1751

)

1 (

04441198971+ 3867

01501198971minus 0031

minus05991198971minus 1751

) (

04441198971+ 3867

01501198971minus 0031

minus05991198971minus 1751

)




my119905

0 (

04441198970

01501198970

minus05991198970

) (

04441198970

01501198970

minus05991198970

)

1 (

04441198971

01501198971

minus05991198971

) (

04441198971

01501198971

minus05991198971

)


119905= 120583119905minus 119904119905119868 =

(0122 0206 0188)1015840 and120593

119905= Σ0

119905= (00006 00149 00050)





6 Conclusion


Appendices

A Proof of (20)


119879minus1into 119906

lowast

119879minus1

1015840Σ1


119879minus1gives

119906lowast

119879minus1

1015840

Σ1


119879minus1=

119868

119886119879minus1

119909119879minus1

+ [Σ0


119868] 119897119879minus1

+120582119879minus1

2[120583119879minus1


119868]

1015840

(Σ1

119879minus1)minus1

times 119868

119886119879minus1

119909119879minus1

+ [Σ0


119868]


+120582119879minus1

2[120583119879minus1


119868]

=1

119886119879minus1

1199092

119879minus1+ [Σ0


119868]1015840

times (Σ1

119879minus1)minus1

[Σ0


119868] 1198972

119879minus1

+ 120582119879minus1

[Σ0


119868]1015840

times (Σ1

119879minus1)minus1

[120583119879minus1


119868] 119897119879minus1

+(120582119879minus1

)2

4[120583119879minus1


119868]1015840

times (Σ1

119879minus1)minus1

[120583119879minus1


119868]

=1

119886119879minus1

1199092

119879minus1+ [(Σ

0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0

119879minus1


1205742

119879minus1] 1198972

119879minus1

+ [(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1


120574119879minus1

]


119897119879minus1

+(120582119879minus1

)2

4120572119879minus1

(A1)



0

119879minus1)1015840

119897119879minus1

+

(120582119879minus1

2)1205831015840



2 [(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840


119879minus1

=2

119886119879minus1

[(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1

119868119909119879minus1

+ 2 [(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1


times [Σ0


119868] 119897119879minus1

+ 120582119879minus1

[(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1



119868]

= 2120574119879minus1

119897119879minus1

119909119879minus1

+ 119902119879minus1

120582119879minus1

119909119879minus1

+ 2 [(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0


119889119879minus1

] 1198972

119879minus1

+ 2120582119879minus1

[(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1


120574119879minus1

] 119897119879minus1

+(120582119879minus1

)2

2120572119879minus1

(A2)

Thus119881lowast

119879minus1(119909119879minus1

119897119879minus1

)

=1

119886119879minus1

1199092


119902119879minus1

119909119879minus1


119909119879minus1

119897119879minus1

+ 120582119879minus1

[119864 (119876119879minus1

) minus (Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1

+119887119879minus1

120574119879minus1

] 119897119879minus1

+ [Var (119876119879minus1

) minus (Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0

119879minus1

+120574119879minus1

119889119879minus1

] 1198972

119879minus1minus

(120582119879minus1

)2

4120572119879minus1

=1

119886119879minus1

1199092


119902119879minus1

119909119879minus1


119909119879minus1

119897119879minus1

+ 1205731119879minus1

119897119879minus1

+ 1205732119879minus1

1198972

119879minus1minus

(120582119879minus1

)2

4120572119879minus1

(A3)


1198681015840

119886119879minus2

(Σ1

119879minus2)minus1

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

=119889119879minus2

119886119879minus2

+ 120574119879minus1

1198681015840(Σ1

119879minus2)minus1

119864 (119876119879minus2

119877119879minus2

)

119886119879minus2


= 120574119879minus2

+ 120574119879minus1

1198681015840(Σ1

119879minus2)minus1

119864 (119876119879minus2

119877119879minus2

)

119886119879minus2


= 120574119879minus2


= 0

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1

[120583119879minus2


119868] =119887119879minus2

119886119879minus2


= 0

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

119868

= 119886119879minus2

120574119879minus2

+ 120574119879minus1

119864 (119876119879minus2

1198771015840

119879minus2) (Σ1

119879minus2)minus1

119868

119886119879minus2

= 119886119879minus2

120574119879minus2

(B1)


119879minus2

1015840Σ1




and 119897119879minus2

119909119879minus2

in 119906lowast

119879minus2

1015840Σ1



119879minus2is

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1 119868

119886119879minus2

=1

119886119879minus2

(B2)


is

1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

= 1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

minus 1205821

119879minus2120574119879minus2

1198681015840(Σ1

119879minus2)minus1

[120583119879minus2


119868]

= 1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

(B3)


is

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]1015840

sdot (Σ1

119879minus2)minus1

sdot [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

= [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]


1205742

119879minus2

(B4)


(1205821

119879minus2)2

4[120583119879minus2


119868]1015840

(Σ1

119879minus2)minus1

[120583119879minus2


119868]

=(1205821

119879minus2)2

4120572119879minus2

(B5)




119879minus2


is

1205821

119879minus2

2

1205831015840

119879minus2(Σ1

119879minus2)minus1

119868

119886119879minus2

=1205821

119879minus2119902119879minus2

2 (B6)



119909119879minus2

is 120574119879minus2


is

1205821

119879minus2

2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

(120583119879minus2


119868)

+1205821

119879minus2

21205831015840

119879minus2(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

(B7)


is

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

= [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)] minus 119886119879minus2

1205742

119879minus2

(B8)


(1205821

119879minus2

2)

2

1205831015840

119879minus2(Σ1

119879minus2)minus1

(120583119879minus2


119868) = (1205821

119879minus2

2)

2

120572119879minus2

(B9)


119879minus2(119909119879minus2

119897119879minus2


119879minus2is

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1 119868

119886119879minus2

=1

119886119879minus2

(B10)



119902119879minus2


119909119879minus2



is

[Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] + 119886

119879minus21205742

119879minus2

minus [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]

= 1205732119879minus2

(B11)


is

minus 1205821

119879minus21205831015840

119879minus2(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)]

= 1205731119879minus2

(B12)


minus1205822

119879minus1

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(B13)

Thus

119881lowast

119879minus2(119909119879minus2

119897119879minus2

) =1

119886119879minus2

1199092


119879minus2119902119879minus2

119909119879minus2


119909119879minus2

119897119879minus2

+ 1205731119879minus2

119897119879minus2

+ 1205732119879minus2

1198972

119879minus2minus

1205822

119879minus1

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(B14)


1198681015840

119886119905minus1

(Σ1

119905minus1)minus1

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

=119889119905minus1

119886119905minus1

+ 120574119905

1198681015840(Σ1

119905minus1)minus1

119864 (119876119905minus1

119877119905minus1

)

119886119905minus1


= 120574119905minus1

+ 120574119905

1198681015840(Σ1

119905minus1)minus1

119864 (119876119905minus1

119877119905minus1

)

119886119905minus1


= 120574119905minus1


= 0

1198681015840

119886119905minus1

(Σ1

119905minus1)minus1

[120583119905minus1


119868] =119887119905minus1

119886119905minus1


= 0

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

119868

= 119886119905minus1

120574119905minus1

+ 120574119905

119864 (119876119905minus1

1198771015840

119905minus1) (Σ1

119905minus1)minus1

119868

119886119905minus1

= 119886119905minus1

120574119905minus1

(C1)


119905minus1

1015840Σ1




and 119897119905minus1

119909119905minus1


is

1198681015840

119886119905minus1

(Σ1

119905minus1)minus1 119868

119886119905minus1

=1

119886119905minus1

(C2)


is

1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

= 1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

minus 1205821

119905minus1120574119905minus1

1198681015840(Σ1

119905minus1)minus1

[120583119905minus1


119868]


= 1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

(C3)


is

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]1015840

sdot (Σ1

119905minus1)minus1

sdot [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

= [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] minus 119886119905minus1

1205742

119905minus1

(C4)


(1205821

119905minus1)2

4[120583119905minus1


119868]1015840

(Σ1

119905minus1)minus1

[120583119905minus1


119868] =(1205821

119905minus1)2

4120572119905minus1

(C5)




119905minus1is


is

1205821

119905minus1

2

1205831015840

119905minus1(Σ1

119905minus1)minus1

119868

119886119905minus1

=1205821

119905minus1119902119905minus1

2 (C6)


119909119905minus1

is 120574119905minus1


is

1205821

119905minus1

2[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1



119868) +1205821

119905minus1

21205831015840

119905minus1(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

(C7)


is

[Σ0

119905minus1+ 120574119905119864(119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

= [Σ0

119905minus1+ 120574119905119864(119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] minus 119886119905minus1

1205742

119905minus1

(C8)


(1205821

119905minus1

2)

2

120583119905minus1

(Σ1

119905minus1)minus1

(120583119905minus1


119868) = (1205821

119905minus1

2)

2

120572119905minus1

(C9)


119905minus1(119909119905minus1

119897119905minus1


119905minus1is 1119886



1

119905minus1119902119905minus1


119905minus1119909119905minus1



is

[Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] + 119886

119905minus11205742

119905minus1

minus [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]

= 1205732119905minus1

(C10)


is

minus 1205821

119905minus11205831015840

119905minus1(Σ1

119905minus1)minus1

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

+ (120582119905minus1

+ 1205731119905) 119864 (119876

119905minus1) = 1205731119905minus1

(C11)


(1205821

119894)2120572119894

Thus

119881lowast

119905minus1(119909119905minus1

119897119905minus1

) =1

119886119905minus1

1199092


119905minus1119902119905minus1

119909119905minus1


119909119905minus1

119897119905minus1

+ 1205731119905minus1

119897119905minus1

+ 1205732119905minus1

1198972

119905minus1minus

1

4

119879minus1

sum119894=119905minus1

(1205821

119894)2

120572119894

(C12)

D Proof of (62)


119879minus1into 119906

lowast

119879minus1

1015840Σ119879minus1

119906lowast


119906lowast

119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1= (1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840

119879minus1)


)minus1

(120593119879minus1

119897119879minus1

+120582119879minus1

2120598119879minus1

)

= 1205931015840


)minus1

120593119879minus1

1198972

119879minus1+ 120582119879minus1

1205981015840

119879minus1


)minus1

120593119879minus1

119897119879minus1

+(120582119879minus1

)2

4119872119879minus1

2 (1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840



119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1

(D1)

Thus

119881lowast

119879minus1(119909119879minus1

119897119879minus1


119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1+ Var (119876

119879minus1) 1198972

119879minus1


119904119879minus1

119909119879minus1

+ 120582119879minus1

119864 (119876119879minus1

) 119897119879minus1

= [Var (119876119879minus1

) minus 1205931015840


)minus1

120593119879minus1

] 1198972

119879minus1

+ 120582119879minus1

[119864 (119876119879minus1

) minus 1205981015840


)minus1

120593119879minus1

]



119904119879minus1

119909119879minus1


)2

4119872119879minus1

= 1205732119879minus1

1198972

119879minus1+ 1205731119879minus1

119897119879minus1


119904119879minus1

119909119879minus1


)2

4119872119879minus1

(D2)



lowast


119906lowast

119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2= (1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840

119879minus2)


)minus1

(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= 1205931015840


)minus1

120593119879minus2

1198972

119879minus2+ 1205821

119879minus21205981015840

119879minus2


)minus1

120593119879minus2

119897119879minus2

+(1205821

119879minus2)2

4119872119879minus2

2 1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840



119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2

(E1)Thus

119881lowast

119879minus2(119909119879minus2

119897119879minus2


119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2

+ [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

= 1205732119879minus2

1198972

119879minus2+ 1205731119879minus2

119897119879minus2

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

minus(1205821

119879minus2)2

4119872119879minus2

(E2)where1205731119879minus2

= (120582119879minus2

+ 1205731119879minus1

) 119864 (119876119879minus2

) minus 1205821

119879minus21205981015840


)minus1

120593119879minus2

1205732119879minus2

= Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2) minus 1205931015840


)minus1

120593119879minus2

(E3)


lowast

119905minus1that

119906lowast

119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1= (1205931015840

119879minus1119897119905minus1

+1205821

119905minus1

21205981015840

119879minus1)


)minus1

(120593119905minus1

119897119905minus1

+1205821

119905minus1

2120598119905minus1

)

= 1205931015840


)minus1

120593119905minus1

1198972

119905minus1+ 1205821

119905minus11205981015840

119879minus1


)minus1

120593119905minus1

119897119905minus1

+(1205821

119905minus1)2

4119872119905minus1

2 (1205931015840

119879minus1119897119905minus1

+1205821

119905minus1

21205981015840



119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1

(F1)

Thus

119881lowast

119905minus1(119909119905minus1

119897119905minus1


119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1

+ [Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] 1198972

119905minus1

+ [120582119905minus1

119864 (119876119905minus1

) + 1205731119905119864 (119876119905minus1

)] 119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

= 1205732119905minus1

1198972

119905minus1+ 1205731119905minus1

119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905minus1

(1205821

119894)2

4119872119894

(F2)

where

1205731119905minus1

= (120582119905minus1

+ 1205731119905) 119864 (119876

119905minus1) minus 1205821

119905minus11205981015840


)minus1

120593119905minus1

1205732119905minus1

= Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1) minus 1205931015840


)minus1

120593119905minus1

(F3)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119879minus1= 120582119879minus1

1205821

119905= 120582119905+ 1205821

119905+1119904119905+1

1205731119879minus1

= 120582119879minus1

[119864 (119876119879minus1

) minus 1205981015840


)minus1

120593119879minus1

] 1205731119905

= (120582119905+ 1205731119905+1

) 119864(119876119905) minus 1205821

1199051205981015840

119905(Σ119905)minus1120593119905

1205732119879minus1

= Var(119876119879minus1

) minus 1205931015840


)minus1

120593119879minus1

1205732119905

= Var(119876119905) + 1205732119905+1

119864(1198762

119905) minus 1205931015840

119905(Σ119905)minus1120593119905




min119906119879minus1

Var119879minus1

(119878119879) minus 120582119879minus1

119864119879minus1

(119878119879)

st 119909119879

= 119904119879minus1

119909119879minus1

+ 1198751015840

119879minus1119906119879minus1

119897119879

= 119876119879minus1

119897119879minus1

119878119879= 119909119879minus 119897119879

(57)


min119906119879minus1

119881119879minus1

(119909119879minus1

119897119879minus1

) (58)

where

119881119879minus1

(119909119879minus1

119897119879minus1

) = 1199061015840


119906119879minus1

minus 2(1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840

119879minus1)119906119879minus1

+ Var (119876119879minus1

) 1198972


119904119879minus1

119909119879minus1

+ 120582119879minus1

119864 (119876119879minus1

) 119897119879minus1

(59)


119889 (119881119879minus1

(119909119879minus1

119897119879minus1

))

119889 (119906119879minus1

)= 2Σ119879minus1

119906119879minus1

minus 2(120593119879minus1

119897119879minus1

+120582119879minus1

2120598119879minus1

)

= 0

(60)

which implies

119906lowast

119879minus1= (Σ119879minus1

)minus1

[120593119879minus1

119897119879minus1

+120582119879minus1

2120598119879minus1

] (61)




119881lowast

119879minus1(119909119879minus1

119897119879minus1

) = 1205732119879minus1

1198972

119879minus1+ 1205731119879minus1

119897119879minus1


119904119879minus1

119909119879minus1


)2

4119872119879minus1

(62)


= 1205981015840


119879minus1120598119879minus1




min Var119879minus2

(119878119879minus1

) minus 120582119879minus2

119864119879minus2

(119878119879minus1

)

+ 119864119879minus2

(119881lowast

119879minus1(119909119879minus1

119897119879minus1

))

st 119909119879minus1

= 119904119879minus2

119909119879minus2

+ 1198751015840

119879minus2119906119879minus2

119897119879minus1

= 119876119879minus2

119897119879minus2

119878119879minus1

= 119909119879minus1


(63)


119864119879minus2

(119881lowast

119879minus1(119909119879minus1

119897119879minus1

)) = 1205732119879minus1

119864 (1198762

119879minus2) 1198972

119879minus2

+ 1205731119879minus1

119864 (119876119879minus2

) 119897119879minus2


119904119879minus1

119904119879minus2

119909119879minus2


119904119879minus1

1205981015840

119879minus2119906119879minus2


)2

4119872119879minus1

(64)


(119881lowast

119879minus1(119909119879minus1

119897119879minus1


min119906119879minus2

119881119879minus2

(119909119879minus2

119897119879minus2

) (65)


(119909119879minus2

119897119879minus2

) = 1199061015840


119906119879minus2

minus 2(1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840

119879minus2)119906119879minus2

+ [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

(66)


119889 (119881119879minus2

(119909119879minus2

119897119879minus2

))

119889 (119906119879minus2

)= 2Σ119879minus2

119906119879minus2

minus 2(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= 0

(67)


Thus

119906lowast

119879minus2= (Σ119879minus2

)minus1

(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= (Σ119879minus2

)minus1

120593119879minus2

119897119879minus2

+1205821

119879minus2

2(Σ119879minus2

)minus1

120598119879minus2

(68)




119881lowast

119879minus2(119909119879minus2

119897119879minus2

) = 1205732119879minus2

1198972

119879minus2+ 1205731119879minus2

119897119879minus2

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

minus(1205821

119879minus2)2

4119872119879minus2

(69)


= 1205981015840


119879minus2120598119879minus2


both (56) and

119881lowast

119905(119909119905 119897119905) = 12057321199051198972

119905+ 1205731119905119897119905minus 1205821

119905119904119905119909119905minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

(70)

hold where 119872119894= 1205981015840

119894Σminus1



119905minus1and



min Var119905minus1

(119878119905) minus 120582119905minus1

119864119905minus1

(119878119905) + 119864119905minus1

(119881lowast

119905(119909119905 119897119905))

st 119909119905= 119904119905minus1

119909119905minus1

+ 1198751015840

119905minus1119906119905minus1

119897119905= 119876119905minus1

119897119905minus1

119878119905= 119909119905minus 119897119905

(71)


119864119905minus1

(119881lowast

119905(119909119905 119897119905)) = 120573

2119905119864 (1198762

119905minus1) 1198972

119905minus1+ 1205731119905119864 (119876119905minus1

) 119897119905

minus 1205821

119905119904119905119904119905minus1

119909119905minus1

minus 1205821

1199051199041199051205981015840

119905minus1119906119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

(72)


(119881lowast



119881119905minus1

(119909119905minus1

119897119905minus1

) (73)

where

119881119905minus1

(119909119905minus1

119897119905minus1

) = 1199061015840


119906119905minus1

minus 2(1205931015840

119905minus1119897119905minus1

+1205821

119905minus1

21205981015840

119905minus1)119906119905minus1

+ [Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] 1198972

119905minus1

+ [120582119905minus1

119864 (119876119905minus1

) + 1205731119905119864 (119876119905minus1

)] 119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

(74)


119889 (119881119905minus1

(119909119905minus1

119897119905minus1

))

119889 (119906119905minus1

)= 2Σ119905minus1

119906119905minus1

minus 2(120593119905minus1

119897119905minus1

+1205821

119905minus1

2120598119905minus1

)

= 0

(75)

which implies

119906lowast


119905minus1120593119905minus1

119897119905minus1

+1205821

119905minus1

2Σminus1

119905minus1120598119905minus1

(76)




119881lowast

119905minus1(119909119905minus1

119897119905minus1

) = 1205732119905minus1

1198972

119905minus1+ 1205731119905minus1

119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905minus1

(1205821

119894)2

4119872119894

(77)







119906lowast

119905=

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (78)






119906lowast

119905=

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (79)


120593119905= 119864 (119875

119905119876119905) minus 119864 (119875

119905) 119864 (119876

119905)

= 119864 ((119877119905minus 119904119905119868)119876119905) minus 119864 (119877

119905minus 119904119905119868) 119864 (119876

119905)

= 119864 (119877119905119876119905) minus 119864 (119877

119905) 119864 (119876

119905)

= Σ0

119905

= 0

(80)



119906lowast

119905=

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (81)




Σminus1

119905120593119905119897119905 (82)



min119906119905

Var119905(119878119905+1

) minus 120582119905119864119905(119878119905+1

)

st 119909119905+1

= 119904119905119909119905+ 1198751015840

119905119906119905

119897119905+1

= 119876119905119897119905

119878119905+1

= 119909119905+1

minus 119897119905+1

(83)


119906119898119910

119905= Σminus1

119905120593119905119897119905+

120582119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (84)




119906lowast

119905= Σminus1

119905120593119905119897119905 119905 = 0 119879 minus 1 (85)






119905


lowast

119905

0 (

11941198970+ 0903119909

0+ 3736

01071198970minus 0066119909

0+ 0203

minus05771198970+ 0163119909

0minus 3939

) (

09031199090+ 3736

minus00661199090+ 0203

01631199090minus 3939

)

1 (

minus03721198971+ 1018119909

1+ 1743

01981198971minus 0060119909

1+ 0095

minus06331198971+ 0042119909

1minus 1838

) (

10181199091+ 1743

minus00601199091+ 0095

00421199091minus 1838

)




my119905

0 (

09031199090+ 1194119897

0+ 3736

minus00661199090+ 0107119897

0+ 0203

01631199090minus 0577119897

0minus 3939

) (

10181199090+ 0449119897

0+ 1743

minus00601199090+ 0150119897

0+ 0095

00421199090minus 0599119897

0minus 1838

)

1 (

10181199091minus 0372119897

1+ 1743

minus00601199091+ 0198119897

1+ 0095

00421199091minus 0633119897

1minus 1838

) (

10181199091+ 0449119897

1+ 1743

minus00601199091+ 0150119897

1+ 0095

00421199091minus 0599119897

1minus 1838

)



119905=

(1162 1246 1228) and

Σ119905= (

00146 00187 00145

00187 00854 00104

00145 00104 00289

) (86)






119905=

05 and 120582119905







my119905

0 (

09031199090+ 1194119897

0

minus00661199090+ 0107119897

0

01631199090minus 0577119897

0

) (

10181199090+ 0449119897

0

minus00601199090+ 0150119897

0

00421199090minus 0599119897

0

)

1 (

10181199091minus 0372119897

1

minus00601199091+ 0198119897

1

00421199091minus 0633119897

1

) (

10181199091+ 0449119897

1

minus00601199091+ 0150119897

1

00421199091minus 0599119897

1

)




my119905

0 (

04441198970+ 7889

01501198970minus 0062

minus05991198970minus 3572

) (

04441198970+ 3867

01501198970minus 0031

minus05991198970minus 1751

)

1 (

04441198971+ 3867

01501198971minus 0031

minus05991198971minus 1751

) (

04441198971+ 3867

01501198971minus 0031

minus05991198971minus 1751

)




my119905

0 (

04441198970

01501198970

minus05991198970

) (

04441198970

01501198970

minus05991198970

)

1 (

04441198971

01501198971

minus05991198971

) (

04441198971

01501198971

minus05991198971

)


119905= 120583119905minus 119904119905119868 =

(0122 0206 0188)1015840 and120593

119905= Σ0

119905= (00006 00149 00050)





6 Conclusion


Appendices

A Proof of (20)


119879minus1into 119906

lowast

119879minus1

1015840Σ1


119879minus1gives

119906lowast

119879minus1

1015840

Σ1


119879minus1=

119868

119886119879minus1

119909119879minus1

+ [Σ0


119868] 119897119879minus1

+120582119879minus1

2[120583119879minus1


119868]

1015840

(Σ1

119879minus1)minus1

times 119868

119886119879minus1

119909119879minus1

+ [Σ0


119868]


+120582119879minus1

2[120583119879minus1


119868]

=1

119886119879minus1

1199092

119879minus1+ [Σ0


119868]1015840

times (Σ1

119879minus1)minus1

[Σ0


119868] 1198972

119879minus1

+ 120582119879minus1

[Σ0


119868]1015840

times (Σ1

119879minus1)minus1

[120583119879minus1


119868] 119897119879minus1

+(120582119879minus1

)2

4[120583119879minus1


119868]1015840

times (Σ1

119879minus1)minus1

[120583119879minus1


119868]

=1

119886119879minus1

1199092

119879minus1+ [(Σ

0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0

119879minus1


1205742

119879minus1] 1198972

119879minus1

+ [(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1


120574119879minus1

]


119897119879minus1

+(120582119879minus1

)2

4120572119879minus1

(A1)



0

119879minus1)1015840

119897119879minus1

+

(120582119879minus1

2)1205831015840



2 [(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840


119879minus1

=2

119886119879minus1

[(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1

119868119909119879minus1

+ 2 [(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1


times [Σ0


119868] 119897119879minus1

+ 120582119879minus1

[(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1



119868]

= 2120574119879minus1

119897119879minus1

119909119879minus1

+ 119902119879minus1

120582119879minus1

119909119879minus1

+ 2 [(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0


119889119879minus1

] 1198972

119879minus1

+ 2120582119879minus1

[(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1


120574119879minus1

] 119897119879minus1

+(120582119879minus1

)2

2120572119879minus1

(A2)

Thus119881lowast

119879minus1(119909119879minus1

119897119879minus1

)

=1

119886119879minus1

1199092


119902119879minus1

119909119879minus1


119909119879minus1

119897119879minus1

+ 120582119879minus1

[119864 (119876119879minus1

) minus (Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1

+119887119879minus1

120574119879minus1

] 119897119879minus1

+ [Var (119876119879minus1

) minus (Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0

119879minus1

+120574119879minus1

119889119879minus1

] 1198972

119879minus1minus

(120582119879minus1

)2

4120572119879minus1

=1

119886119879minus1

1199092


119902119879minus1

119909119879minus1


119909119879minus1

119897119879minus1

+ 1205731119879minus1

119897119879minus1

+ 1205732119879minus1

1198972

119879minus1minus

(120582119879minus1

)2

4120572119879minus1

(A3)


1198681015840

119886119879minus2

(Σ1

119879minus2)minus1

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

=119889119879minus2

119886119879minus2

+ 120574119879minus1

1198681015840(Σ1

119879minus2)minus1

119864 (119876119879minus2

119877119879minus2

)

119886119879minus2


= 120574119879minus2

+ 120574119879minus1

1198681015840(Σ1

119879minus2)minus1

119864 (119876119879minus2

119877119879minus2

)

119886119879minus2


= 120574119879minus2


= 0

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1

[120583119879minus2


119868] =119887119879minus2

119886119879minus2


= 0

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

119868

= 119886119879minus2

120574119879minus2

+ 120574119879minus1

119864 (119876119879minus2

1198771015840

119879minus2) (Σ1

119879minus2)minus1

119868

119886119879minus2

= 119886119879minus2

120574119879minus2

(B1)


119879minus2

1015840Σ1




and 119897119879minus2

119909119879minus2

in 119906lowast

119879minus2

1015840Σ1



119879minus2is

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1 119868

119886119879minus2

=1

119886119879minus2

(B2)


is

1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

= 1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

minus 1205821

119879minus2120574119879minus2

1198681015840(Σ1

119879minus2)minus1

[120583119879minus2


119868]

= 1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

(B3)


is

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]1015840

sdot (Σ1

119879minus2)minus1

sdot [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

= [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]


1205742

119879minus2

(B4)


(1205821

119879minus2)2

4[120583119879minus2


119868]1015840

(Σ1

119879minus2)minus1

[120583119879minus2


119868]

=(1205821

119879minus2)2

4120572119879minus2

(B5)




119879minus2


is

1205821

119879minus2

2

1205831015840

119879minus2(Σ1

119879minus2)minus1

119868

119886119879minus2

=1205821

119879minus2119902119879minus2

2 (B6)



119909119879minus2

is 120574119879minus2


is

1205821

119879minus2

2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

(120583119879minus2


119868)

+1205821

119879minus2

21205831015840

119879minus2(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

(B7)


is

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

= [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)] minus 119886119879minus2

1205742

119879minus2

(B8)


(1205821

119879minus2

2)

2

1205831015840

119879minus2(Σ1

119879minus2)minus1

(120583119879minus2


119868) = (1205821

119879minus2

2)

2

120572119879minus2

(B9)


119879minus2(119909119879minus2

119897119879minus2


119879minus2is

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1 119868

119886119879minus2

=1

119886119879minus2

(B10)



119902119879minus2


119909119879minus2



is

[Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] + 119886

119879minus21205742

119879minus2

minus [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]

= 1205732119879minus2

(B11)


is

minus 1205821

119879minus21205831015840

119879minus2(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)]

= 1205731119879minus2

(B12)


minus1205822

119879minus1

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(B13)

Thus

119881lowast

119879minus2(119909119879minus2

119897119879minus2

) =1

119886119879minus2

1199092


119879minus2119902119879minus2

119909119879minus2


119909119879minus2

119897119879minus2

+ 1205731119879minus2

119897119879minus2

+ 1205732119879minus2

1198972

119879minus2minus

1205822

119879minus1

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(B14)


1198681015840

119886119905minus1

(Σ1

119905minus1)minus1

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

=119889119905minus1

119886119905minus1

+ 120574119905

1198681015840(Σ1

119905minus1)minus1

119864 (119876119905minus1

119877119905minus1

)

119886119905minus1


= 120574119905minus1

+ 120574119905

1198681015840(Σ1

119905minus1)minus1

119864 (119876119905minus1

119877119905minus1

)

119886119905minus1


= 120574119905minus1


= 0

1198681015840

119886119905minus1

(Σ1

119905minus1)minus1

[120583119905minus1


119868] =119887119905minus1

119886119905minus1


= 0

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

119868

= 119886119905minus1

120574119905minus1

+ 120574119905

119864 (119876119905minus1

1198771015840

119905minus1) (Σ1

119905minus1)minus1

119868

119886119905minus1

= 119886119905minus1

120574119905minus1

(C1)


119905minus1

1015840Σ1




and 119897119905minus1

119909119905minus1


is

1198681015840

119886119905minus1

(Σ1

119905minus1)minus1 119868

119886119905minus1

=1

119886119905minus1

(C2)


is

1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

= 1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

minus 1205821

119905minus1120574119905minus1

1198681015840(Σ1

119905minus1)minus1

[120583119905minus1


119868]


= 1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

(C3)


is

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]1015840

sdot (Σ1

119905minus1)minus1

sdot [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

= [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] minus 119886119905minus1

1205742

119905minus1

(C4)


(1205821

119905minus1)2

4[120583119905minus1


119868]1015840

(Σ1

119905minus1)minus1

[120583119905minus1


119868] =(1205821

119905minus1)2

4120572119905minus1

(C5)




119905minus1is


is

1205821

119905minus1

2

1205831015840

119905minus1(Σ1

119905minus1)minus1

119868

119886119905minus1

=1205821

119905minus1119902119905minus1

2 (C6)


119909119905minus1

is 120574119905minus1


is

1205821

119905minus1

2[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1



119868) +1205821

119905minus1

21205831015840

119905minus1(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

(C7)


is

[Σ0

119905minus1+ 120574119905119864(119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

= [Σ0

119905minus1+ 120574119905119864(119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] minus 119886119905minus1

1205742

119905minus1

(C8)


(1205821

119905minus1

2)

2

120583119905minus1

(Σ1

119905minus1)minus1

(120583119905minus1


119868) = (1205821

119905minus1

2)

2

120572119905minus1

(C9)


119905minus1(119909119905minus1

119897119905minus1


119905minus1is 1119886



1

119905minus1119902119905minus1


119905minus1119909119905minus1



is

[Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] + 119886

119905minus11205742

119905minus1

minus [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]

= 1205732119905minus1

(C10)


is

minus 1205821

119905minus11205831015840

119905minus1(Σ1

119905minus1)minus1

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

+ (120582119905minus1

+ 1205731119905) 119864 (119876

119905minus1) = 1205731119905minus1

(C11)


(1205821

119894)2120572119894

Thus

119881lowast

119905minus1(119909119905minus1

119897119905minus1

) =1

119886119905minus1

1199092


119905minus1119902119905minus1

119909119905minus1


119909119905minus1

119897119905minus1

+ 1205731119905minus1

119897119905minus1

+ 1205732119905minus1

1198972

119905minus1minus

1

4

119879minus1

sum119894=119905minus1

(1205821

119894)2

120572119894

(C12)

D Proof of (62)


119879minus1into 119906

lowast

119879minus1

1015840Σ119879minus1

119906lowast


119906lowast

119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1= (1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840

119879minus1)


)minus1

(120593119879minus1

119897119879minus1

+120582119879minus1

2120598119879minus1

)

= 1205931015840


)minus1

120593119879minus1

1198972

119879minus1+ 120582119879minus1

1205981015840

119879minus1


)minus1

120593119879minus1

119897119879minus1

+(120582119879minus1

)2

4119872119879minus1

2 (1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840



119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1

(D1)

Thus

119881lowast

119879minus1(119909119879minus1

119897119879minus1


119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1+ Var (119876

119879minus1) 1198972

119879minus1


119904119879minus1

119909119879minus1

+ 120582119879minus1

119864 (119876119879minus1

) 119897119879minus1

= [Var (119876119879minus1

) minus 1205931015840


)minus1

120593119879minus1

] 1198972

119879minus1

+ 120582119879minus1

[119864 (119876119879minus1

) minus 1205981015840


)minus1

120593119879minus1

]



119904119879minus1

119909119879minus1


)2

4119872119879minus1

= 1205732119879minus1

1198972

119879minus1+ 1205731119879minus1

119897119879minus1


119904119879minus1

119909119879minus1


)2

4119872119879minus1

(D2)



lowast


119906lowast

119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2= (1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840

119879minus2)


)minus1

(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= 1205931015840


)minus1

120593119879minus2

1198972

119879minus2+ 1205821

119879minus21205981015840

119879minus2


)minus1

120593119879minus2

119897119879minus2

+(1205821

119879minus2)2

4119872119879minus2

2 1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840



119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2

(E1)Thus

119881lowast

119879minus2(119909119879minus2

119897119879minus2


119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2

+ [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

= 1205732119879minus2

1198972

119879minus2+ 1205731119879minus2

119897119879minus2

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

minus(1205821

119879minus2)2

4119872119879minus2

(E2)where1205731119879minus2

= (120582119879minus2

+ 1205731119879minus1

) 119864 (119876119879minus2

) minus 1205821

119879minus21205981015840


)minus1

120593119879minus2

1205732119879minus2

= Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2) minus 1205931015840


)minus1

120593119879minus2

(E3)


lowast

119905minus1that

119906lowast

119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1= (1205931015840

119879minus1119897119905minus1

+1205821

119905minus1

21205981015840

119879minus1)


)minus1

(120593119905minus1

119897119905minus1

+1205821

119905minus1

2120598119905minus1

)

= 1205931015840


)minus1

120593119905minus1

1198972

119905minus1+ 1205821

119905minus11205981015840

119879minus1


)minus1

120593119905minus1

119897119905minus1

+(1205821

119905minus1)2

4119872119905minus1

2 (1205931015840

119879minus1119897119905minus1

+1205821

119905minus1

21205981015840



119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1

(F1)

Thus

119881lowast

119905minus1(119909119905minus1

119897119905minus1


119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1

+ [Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] 1198972

119905minus1

+ [120582119905minus1

119864 (119876119905minus1

) + 1205731119905119864 (119876119905minus1

)] 119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

= 1205732119905minus1

1198972

119905minus1+ 1205731119905minus1

119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905minus1

(1205821

119894)2

4119872119894

(F2)

where

1205731119905minus1

= (120582119905minus1

+ 1205731119905) 119864 (119876

119905minus1) minus 1205821

119905minus11205981015840


)minus1

120593119905minus1

1205732119905minus1

= Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1) minus 1205931015840


)minus1

120593119905minus1

(F3)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Thus

119906lowast

119879minus2= (Σ119879minus2

)minus1

(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= (Σ119879minus2

)minus1

120593119879minus2

119897119879minus2

+1205821

119879minus2

2(Σ119879minus2

)minus1

120598119879minus2

(68)




119881lowast

119879minus2(119909119879minus2

119897119879minus2

) = 1205732119879minus2

1198972

119879minus2+ 1205731119879minus2

119897119879minus2

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

minus(1205821

119879minus2)2

4119872119879minus2

(69)


= 1205981015840


119879minus2120598119879minus2


both (56) and

119881lowast

119905(119909119905 119897119905) = 12057321199051198972

119905+ 1205731119905119897119905minus 1205821

119905119904119905119909119905minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

(70)

hold where 119872119894= 1205981015840

119894Σminus1



119905minus1and



min Var119905minus1

(119878119905) minus 120582119905minus1

119864119905minus1

(119878119905) + 119864119905minus1

(119881lowast

119905(119909119905 119897119905))

st 119909119905= 119904119905minus1

119909119905minus1

+ 1198751015840

119905minus1119906119905minus1

119897119905= 119876119905minus1

119897119905minus1

119878119905= 119909119905minus 119897119905

(71)


119864119905minus1

(119881lowast

119905(119909119905 119897119905)) = 120573

2119905119864 (1198762

119905minus1) 1198972

119905minus1+ 1205731119905119864 (119876119905minus1

) 119897119905

minus 1205821

119905119904119905119904119905minus1

119909119905minus1

minus 1205821

1199051199041199051205981015840

119905minus1119906119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

(72)


(119881lowast



119881119905minus1

(119909119905minus1

119897119905minus1

) (73)

where

119881119905minus1

(119909119905minus1

119897119905minus1

) = 1199061015840


119906119905minus1

minus 2(1205931015840

119905minus1119897119905minus1

+1205821

119905minus1

21205981015840

119905minus1)119906119905minus1

+ [Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] 1198972

119905minus1

+ [120582119905minus1

119864 (119876119905minus1

) + 1205731119905119864 (119876119905minus1

)] 119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

(74)


119889 (119881119905minus1

(119909119905minus1

119897119905minus1

))

119889 (119906119905minus1

)= 2Σ119905minus1

119906119905minus1

minus 2(120593119905minus1

119897119905minus1

+1205821

119905minus1

2120598119905minus1

)

= 0

(75)

which implies

119906lowast


119905minus1120593119905minus1

119897119905minus1

+1205821

119905minus1

2Σminus1

119905minus1120598119905minus1

(76)




119881lowast

119905minus1(119909119905minus1

119897119905minus1

) = 1205732119905minus1

1198972

119905minus1+ 1205731119905minus1

119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905minus1

(1205821

119894)2

4119872119894

(77)







119906lowast

119905=

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (78)






119906lowast

119905=

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (79)


120593119905= 119864 (119875

119905119876119905) minus 119864 (119875

119905) 119864 (119876

119905)

= 119864 ((119877119905minus 119904119905119868)119876119905) minus 119864 (119877

119905minus 119904119905119868) 119864 (119876

119905)

= 119864 (119877119905119876119905) minus 119864 (119877

119905) 119864 (119876

119905)

= Σ0

119905

= 0

(80)



119906lowast

119905=

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (81)




Σminus1

119905120593119905119897119905 (82)



min119906119905

Var119905(119878119905+1

) minus 120582119905119864119905(119878119905+1

)

st 119909119905+1

= 119904119905119909119905+ 1198751015840

119905119906119905

119897119905+1

= 119876119905119897119905

119878119905+1

= 119909119905+1

minus 119897119905+1

(83)


119906119898119910

119905= Σminus1

119905120593119905119897119905+

120582119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (84)




119906lowast

119905= Σminus1

119905120593119905119897119905 119905 = 0 119879 minus 1 (85)






119905


lowast

119905

0 (

11941198970+ 0903119909

0+ 3736

01071198970minus 0066119909

0+ 0203

minus05771198970+ 0163119909

0minus 3939

) (

09031199090+ 3736

minus00661199090+ 0203

01631199090minus 3939

)

1 (

minus03721198971+ 1018119909

1+ 1743

01981198971minus 0060119909

1+ 0095

minus06331198971+ 0042119909

1minus 1838

) (

10181199091+ 1743

minus00601199091+ 0095

00421199091minus 1838

)




my119905

0 (

09031199090+ 1194119897

0+ 3736

minus00661199090+ 0107119897

0+ 0203

01631199090minus 0577119897

0minus 3939

) (

10181199090+ 0449119897

0+ 1743

minus00601199090+ 0150119897

0+ 0095

00421199090minus 0599119897

0minus 1838

)

1 (

10181199091minus 0372119897

1+ 1743

minus00601199091+ 0198119897

1+ 0095

00421199091minus 0633119897

1minus 1838

) (

10181199091+ 0449119897

1+ 1743

minus00601199091+ 0150119897

1+ 0095

00421199091minus 0599119897

1minus 1838

)



119905=

(1162 1246 1228) and

Σ119905= (

00146 00187 00145

00187 00854 00104

00145 00104 00289

) (86)






119905=

05 and 120582119905







my119905

0 (

09031199090+ 1194119897

0

minus00661199090+ 0107119897

0

01631199090minus 0577119897

0

) (

10181199090+ 0449119897

0

minus00601199090+ 0150119897

0

00421199090minus 0599119897

0

)

1 (

10181199091minus 0372119897

1

minus00601199091+ 0198119897

1

00421199091minus 0633119897

1

) (

10181199091+ 0449119897

1

minus00601199091+ 0150119897

1

00421199091minus 0599119897

1

)




my119905

0 (

04441198970+ 7889

01501198970minus 0062

minus05991198970minus 3572

) (

04441198970+ 3867

01501198970minus 0031

minus05991198970minus 1751

)

1 (

04441198971+ 3867

01501198971minus 0031

minus05991198971minus 1751

) (

04441198971+ 3867

01501198971minus 0031

minus05991198971minus 1751

)




my119905

0 (

04441198970

01501198970

minus05991198970

) (

04441198970

01501198970

minus05991198970

)

1 (

04441198971

01501198971

minus05991198971

) (

04441198971

01501198971

minus05991198971

)


119905= 120583119905minus 119904119905119868 =

(0122 0206 0188)1015840 and120593

119905= Σ0

119905= (00006 00149 00050)





6 Conclusion


Appendices

A Proof of (20)


119879minus1into 119906

lowast

119879minus1

1015840Σ1


119879minus1gives

119906lowast

119879minus1

1015840

Σ1


119879minus1=

119868

119886119879minus1

119909119879minus1

+ [Σ0


119868] 119897119879minus1

+120582119879minus1

2[120583119879minus1


119868]

1015840

(Σ1

119879minus1)minus1

times 119868

119886119879minus1

119909119879minus1

+ [Σ0


119868]


+120582119879minus1

2[120583119879minus1


119868]

=1

119886119879minus1

1199092

119879minus1+ [Σ0


119868]1015840

times (Σ1

119879minus1)minus1

[Σ0


119868] 1198972

119879minus1

+ 120582119879minus1

[Σ0


119868]1015840

times (Σ1

119879minus1)minus1

[120583119879minus1


119868] 119897119879minus1

+(120582119879minus1

)2

4[120583119879minus1


119868]1015840

times (Σ1

119879minus1)minus1

[120583119879minus1


119868]

=1

119886119879minus1

1199092

119879minus1+ [(Σ

0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0

119879minus1


1205742

119879minus1] 1198972

119879minus1

+ [(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1


120574119879minus1

]


119897119879minus1

+(120582119879minus1

)2

4120572119879minus1

(A1)



0

119879minus1)1015840

119897119879minus1

+

(120582119879minus1

2)1205831015840



2 [(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840


119879minus1

=2

119886119879minus1

[(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1

119868119909119879minus1

+ 2 [(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1


times [Σ0


119868] 119897119879minus1

+ 120582119879minus1

[(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1



119868]

= 2120574119879minus1

119897119879minus1

119909119879minus1

+ 119902119879minus1

120582119879minus1

119909119879minus1

+ 2 [(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0


119889119879minus1

] 1198972

119879minus1

+ 2120582119879minus1

[(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1


120574119879minus1

] 119897119879minus1

+(120582119879minus1

)2

2120572119879minus1

(A2)

Thus119881lowast

119879minus1(119909119879minus1

119897119879minus1

)

=1

119886119879minus1

1199092


119902119879minus1

119909119879minus1


119909119879minus1

119897119879minus1

+ 120582119879minus1

[119864 (119876119879minus1

) minus (Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1

+119887119879minus1

120574119879minus1

] 119897119879minus1

+ [Var (119876119879minus1

) minus (Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0

119879minus1

+120574119879minus1

119889119879minus1

] 1198972

119879minus1minus

(120582119879minus1

)2

4120572119879minus1

=1

119886119879minus1

1199092


119902119879minus1

119909119879minus1


119909119879minus1

119897119879minus1

+ 1205731119879minus1

119897119879minus1

+ 1205732119879minus1

1198972

119879minus1minus

(120582119879minus1

)2

4120572119879minus1

(A3)


1198681015840

119886119879minus2

(Σ1

119879minus2)minus1

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

=119889119879minus2

119886119879minus2

+ 120574119879minus1

1198681015840(Σ1

119879minus2)minus1

119864 (119876119879minus2

119877119879minus2

)

119886119879minus2


= 120574119879minus2

+ 120574119879minus1

1198681015840(Σ1

119879minus2)minus1

119864 (119876119879minus2

119877119879minus2

)

119886119879minus2


= 120574119879minus2


= 0

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1

[120583119879minus2


119868] =119887119879minus2

119886119879minus2


= 0

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

119868

= 119886119879minus2

120574119879minus2

+ 120574119879minus1

119864 (119876119879minus2

1198771015840

119879minus2) (Σ1

119879minus2)minus1

119868

119886119879minus2

= 119886119879minus2

120574119879minus2

(B1)


119879minus2

1015840Σ1




and 119897119879minus2

119909119879minus2

in 119906lowast

119879minus2

1015840Σ1



119879minus2is

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1 119868

119886119879minus2

=1

119886119879minus2

(B2)


is

1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

= 1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

minus 1205821

119879minus2120574119879minus2

1198681015840(Σ1

119879minus2)minus1

[120583119879minus2


119868]

= 1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

(B3)


is

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]1015840

sdot (Σ1

119879minus2)minus1

sdot [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

= [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]


1205742

119879minus2

(B4)


(1205821

119879minus2)2

4[120583119879minus2


119868]1015840

(Σ1

119879minus2)minus1

[120583119879minus2


119868]

=(1205821

119879minus2)2

4120572119879minus2

(B5)




119879minus2


is

1205821

119879minus2

2

1205831015840

119879minus2(Σ1

119879minus2)minus1

119868

119886119879minus2

=1205821

119879minus2119902119879minus2

2 (B6)



119909119879minus2

is 120574119879minus2


is

1205821

119879minus2

2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

(120583119879minus2


119868)

+1205821

119879minus2

21205831015840

119879minus2(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

(B7)


is

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

= [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)] minus 119886119879minus2

1205742

119879minus2

(B8)


(1205821

119879minus2

2)

2

1205831015840

119879minus2(Σ1

119879minus2)minus1

(120583119879minus2


119868) = (1205821

119879minus2

2)

2

120572119879minus2

(B9)


119879minus2(119909119879minus2

119897119879minus2


119879minus2is

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1 119868

119886119879minus2

=1

119886119879minus2

(B10)



119902119879minus2


119909119879minus2



is

[Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] + 119886

119879minus21205742

119879minus2

minus [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]

= 1205732119879minus2

(B11)


is

minus 1205821

119879minus21205831015840

119879minus2(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)]

= 1205731119879minus2

(B12)


minus1205822

119879minus1

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(B13)

Thus

119881lowast

119879minus2(119909119879minus2

119897119879minus2

) =1

119886119879minus2

1199092


119879minus2119902119879minus2

119909119879minus2


119909119879minus2

119897119879minus2

+ 1205731119879minus2

119897119879minus2

+ 1205732119879minus2

1198972

119879minus2minus

1205822

119879minus1

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(B14)


1198681015840

119886119905minus1

(Σ1

119905minus1)minus1

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

=119889119905minus1

119886119905minus1

+ 120574119905

1198681015840(Σ1

119905minus1)minus1

119864 (119876119905minus1

119877119905minus1

)

119886119905minus1


= 120574119905minus1

+ 120574119905

1198681015840(Σ1

119905minus1)minus1

119864 (119876119905minus1

119877119905minus1

)

119886119905minus1


= 120574119905minus1


= 0

1198681015840

119886119905minus1

(Σ1

119905minus1)minus1

[120583119905minus1


119868] =119887119905minus1

119886119905minus1


= 0

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

119868

= 119886119905minus1

120574119905minus1

+ 120574119905

119864 (119876119905minus1

1198771015840

119905minus1) (Σ1

119905minus1)minus1

119868

119886119905minus1

= 119886119905minus1

120574119905minus1

(C1)


119905minus1

1015840Σ1




and 119897119905minus1

119909119905minus1


is

1198681015840

119886119905minus1

(Σ1

119905minus1)minus1 119868

119886119905minus1

=1

119886119905minus1

(C2)


is

1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

= 1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

minus 1205821

119905minus1120574119905minus1

1198681015840(Σ1

119905minus1)minus1

[120583119905minus1


119868]


= 1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

(C3)


is

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]1015840

sdot (Σ1

119905minus1)minus1

sdot [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

= [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] minus 119886119905minus1

1205742

119905minus1

(C4)


(1205821

119905minus1)2

4[120583119905minus1


119868]1015840

(Σ1

119905minus1)minus1

[120583119905minus1


119868] =(1205821

119905minus1)2

4120572119905minus1

(C5)




119905minus1is


is

1205821

119905minus1

2

1205831015840

119905minus1(Σ1

119905minus1)minus1

119868

119886119905minus1

=1205821

119905minus1119902119905minus1

2 (C6)


119909119905minus1

is 120574119905minus1


is

1205821

119905minus1

2[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1



119868) +1205821

119905minus1

21205831015840

119905minus1(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

(C7)


is

[Σ0

119905minus1+ 120574119905119864(119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

= [Σ0

119905minus1+ 120574119905119864(119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] minus 119886119905minus1

1205742

119905minus1

(C8)


(1205821

119905minus1

2)

2

120583119905minus1

(Σ1

119905minus1)minus1

(120583119905minus1


119868) = (1205821

119905minus1

2)

2

120572119905minus1

(C9)


119905minus1(119909119905minus1

119897119905minus1


119905minus1is 1119886



1

119905minus1119902119905minus1


119905minus1119909119905minus1



is

[Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] + 119886

119905minus11205742

119905minus1

minus [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]

= 1205732119905minus1

(C10)


is

minus 1205821

119905minus11205831015840

119905minus1(Σ1

119905minus1)minus1

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

+ (120582119905minus1

+ 1205731119905) 119864 (119876

119905minus1) = 1205731119905minus1

(C11)


(1205821

119894)2120572119894

Thus

119881lowast

119905minus1(119909119905minus1

119897119905minus1

) =1

119886119905minus1

1199092


119905minus1119902119905minus1

119909119905minus1


119909119905minus1

119897119905minus1

+ 1205731119905minus1

119897119905minus1

+ 1205732119905minus1

1198972

119905minus1minus

1

4

119879minus1

sum119894=119905minus1

(1205821

119894)2

120572119894

(C12)

D Proof of (62)


119879minus1into 119906

lowast

119879minus1

1015840Σ119879minus1

119906lowast


119906lowast

119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1= (1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840

119879minus1)


)minus1

(120593119879minus1

119897119879minus1

+120582119879minus1

2120598119879minus1

)

= 1205931015840


)minus1

120593119879minus1

1198972

119879minus1+ 120582119879minus1

1205981015840

119879minus1


)minus1

120593119879minus1

119897119879minus1

+(120582119879minus1

)2

4119872119879minus1

2 (1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840



119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1

(D1)

Thus

119881lowast

119879minus1(119909119879minus1

119897119879minus1


119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1+ Var (119876

119879minus1) 1198972

119879minus1


119904119879minus1

119909119879minus1

+ 120582119879minus1

119864 (119876119879minus1

) 119897119879minus1

= [Var (119876119879minus1

) minus 1205931015840


)minus1

120593119879minus1

] 1198972

119879minus1

+ 120582119879minus1

[119864 (119876119879minus1

) minus 1205981015840


)minus1

120593119879minus1

]



119904119879minus1

119909119879minus1


)2

4119872119879minus1

= 1205732119879minus1

1198972

119879minus1+ 1205731119879minus1

119897119879minus1


119904119879minus1

119909119879minus1


)2

4119872119879minus1

(D2)



lowast


119906lowast

119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2= (1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840

119879minus2)


)minus1

(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= 1205931015840


)minus1

120593119879minus2

1198972

119879minus2+ 1205821

119879minus21205981015840

119879minus2


)minus1

120593119879minus2

119897119879minus2

+(1205821

119879minus2)2

4119872119879minus2

2 1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840



119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2

(E1)Thus

119881lowast

119879minus2(119909119879minus2

119897119879minus2


119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2

+ [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

= 1205732119879minus2

1198972

119879minus2+ 1205731119879minus2

119897119879minus2

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

minus(1205821

119879minus2)2

4119872119879minus2

(E2)where1205731119879minus2

= (120582119879minus2

+ 1205731119879minus1

) 119864 (119876119879minus2

) minus 1205821

119879minus21205981015840


)minus1

120593119879minus2

1205732119879minus2

= Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2) minus 1205931015840


)minus1

120593119879minus2

(E3)


lowast

119905minus1that

119906lowast

119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1= (1205931015840

119879minus1119897119905minus1

+1205821

119905minus1

21205981015840

119879minus1)


)minus1

(120593119905minus1

119897119905minus1

+1205821

119905minus1

2120598119905minus1

)

= 1205931015840


)minus1

120593119905minus1

1198972

119905minus1+ 1205821

119905minus11205981015840

119879minus1


)minus1

120593119905minus1

119897119905minus1

+(1205821

119905minus1)2

4119872119905minus1

2 (1205931015840

119879minus1119897119905minus1

+1205821

119905minus1

21205981015840



119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1

(F1)

Thus

119881lowast

119905minus1(119909119905minus1

119897119905minus1


119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1

+ [Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] 1198972

119905minus1

+ [120582119905minus1

119864 (119876119905minus1

) + 1205731119905119864 (119876119905minus1

)] 119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

= 1205732119905minus1

1198972

119905minus1+ 1205731119905minus1

119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905minus1

(1205821

119894)2

4119872119894

(F2)

where

1205731119905minus1

= (120582119905minus1

+ 1205731119905) 119864 (119876

119905minus1) minus 1205821

119905minus11205981015840


)minus1

120593119905minus1

1205732119905minus1

= Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1) minus 1205931015840


)minus1

120593119905minus1

(F3)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119906lowast

119905=

1205821

119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (81)




Σminus1

119905120593119905119897119905 (82)



min119906119905

Var119905(119878119905+1

) minus 120582119905119864119905(119878119905+1

)

st 119909119905+1

= 119904119905119909119905+ 1198751015840

119905119906119905

119897119905+1

= 119876119905119897119905

119878119905+1

= 119909119905+1

minus 119897119905+1

(83)


119906119898119910

119905= Σminus1

119905120593119905119897119905+

120582119905

2Σminus1

119905120598119905 119905 = 0 119879 minus 1 (84)




119906lowast

119905= Σminus1

119905120593119905119897119905 119905 = 0 119879 minus 1 (85)






119905


lowast

119905

0 (

11941198970+ 0903119909

0+ 3736

01071198970minus 0066119909

0+ 0203

minus05771198970+ 0163119909

0minus 3939

) (

09031199090+ 3736

minus00661199090+ 0203

01631199090minus 3939

)

1 (

minus03721198971+ 1018119909

1+ 1743

01981198971minus 0060119909

1+ 0095

minus06331198971+ 0042119909

1minus 1838

) (

10181199091+ 1743

minus00601199091+ 0095

00421199091minus 1838

)




my119905

0 (

09031199090+ 1194119897

0+ 3736

minus00661199090+ 0107119897

0+ 0203

01631199090minus 0577119897

0minus 3939

) (

10181199090+ 0449119897

0+ 1743

minus00601199090+ 0150119897

0+ 0095

00421199090minus 0599119897

0minus 1838

)

1 (

10181199091minus 0372119897

1+ 1743

minus00601199091+ 0198119897

1+ 0095

00421199091minus 0633119897

1minus 1838

) (

10181199091+ 0449119897

1+ 1743

minus00601199091+ 0150119897

1+ 0095

00421199091minus 0599119897

1minus 1838

)



119905=

(1162 1246 1228) and

Σ119905= (

00146 00187 00145

00187 00854 00104

00145 00104 00289

) (86)






119905=

05 and 120582119905







my119905

0 (

09031199090+ 1194119897

0

minus00661199090+ 0107119897

0

01631199090minus 0577119897

0

) (

10181199090+ 0449119897

0

minus00601199090+ 0150119897

0

00421199090minus 0599119897

0

)

1 (

10181199091minus 0372119897

1

minus00601199091+ 0198119897

1

00421199091minus 0633119897

1

) (

10181199091+ 0449119897

1

minus00601199091+ 0150119897

1

00421199091minus 0599119897

1

)




my119905

0 (

04441198970+ 7889

01501198970minus 0062

minus05991198970minus 3572

) (

04441198970+ 3867

01501198970minus 0031

minus05991198970minus 1751

)

1 (

04441198971+ 3867

01501198971minus 0031

minus05991198971minus 1751

) (

04441198971+ 3867

01501198971minus 0031

minus05991198971minus 1751

)




my119905

0 (

04441198970

01501198970

minus05991198970

) (

04441198970

01501198970

minus05991198970

)

1 (

04441198971

01501198971

minus05991198971

) (

04441198971

01501198971

minus05991198971

)


119905= 120583119905minus 119904119905119868 =

(0122 0206 0188)1015840 and120593

119905= Σ0

119905= (00006 00149 00050)





6 Conclusion


Appendices

A Proof of (20)


119879minus1into 119906

lowast

119879minus1

1015840Σ1


119879minus1gives

119906lowast

119879minus1

1015840

Σ1


119879minus1=

119868

119886119879minus1

119909119879minus1

+ [Σ0


119868] 119897119879minus1

+120582119879minus1

2[120583119879minus1


119868]

1015840

(Σ1

119879minus1)minus1

times 119868

119886119879minus1

119909119879minus1

+ [Σ0


119868]


+120582119879minus1

2[120583119879minus1


119868]

=1

119886119879minus1

1199092

119879minus1+ [Σ0


119868]1015840

times (Σ1

119879minus1)minus1

[Σ0


119868] 1198972

119879minus1

+ 120582119879minus1

[Σ0


119868]1015840

times (Σ1

119879minus1)minus1

[120583119879minus1


119868] 119897119879minus1

+(120582119879minus1

)2

4[120583119879minus1


119868]1015840

times (Σ1

119879minus1)minus1

[120583119879minus1


119868]

=1

119886119879minus1

1199092

119879minus1+ [(Σ

0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0

119879minus1


1205742

119879minus1] 1198972

119879minus1

+ [(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1


120574119879minus1

]


119897119879minus1

+(120582119879minus1

)2

4120572119879minus1

(A1)



0

119879minus1)1015840

119897119879minus1

+

(120582119879minus1

2)1205831015840



2 [(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840


119879minus1

=2

119886119879minus1

[(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1

119868119909119879minus1

+ 2 [(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1


times [Σ0


119868] 119897119879minus1

+ 120582119879minus1

[(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1



119868]

= 2120574119879minus1

119897119879minus1

119909119879minus1

+ 119902119879minus1

120582119879minus1

119909119879minus1

+ 2 [(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0


119889119879minus1

] 1198972

119879minus1

+ 2120582119879minus1

[(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1


120574119879minus1

] 119897119879minus1

+(120582119879minus1

)2

2120572119879minus1

(A2)

Thus119881lowast

119879minus1(119909119879minus1

119897119879minus1

)

=1

119886119879minus1

1199092


119902119879minus1

119909119879minus1


119909119879minus1

119897119879minus1

+ 120582119879minus1

[119864 (119876119879minus1

) minus (Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1

+119887119879minus1

120574119879minus1

] 119897119879minus1

+ [Var (119876119879minus1

) minus (Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0

119879minus1

+120574119879minus1

119889119879minus1

] 1198972

119879minus1minus

(120582119879minus1

)2

4120572119879minus1

=1

119886119879minus1

1199092


119902119879minus1

119909119879minus1


119909119879minus1

119897119879minus1

+ 1205731119879minus1

119897119879minus1

+ 1205732119879minus1

1198972

119879minus1minus

(120582119879minus1

)2

4120572119879minus1

(A3)


1198681015840

119886119879minus2

(Σ1

119879minus2)minus1

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

=119889119879minus2

119886119879minus2

+ 120574119879minus1

1198681015840(Σ1

119879minus2)minus1

119864 (119876119879minus2

119877119879minus2

)

119886119879minus2


= 120574119879minus2

+ 120574119879minus1

1198681015840(Σ1

119879minus2)minus1

119864 (119876119879minus2

119877119879minus2

)

119886119879minus2


= 120574119879minus2


= 0

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1

[120583119879minus2


119868] =119887119879minus2

119886119879minus2


= 0

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

119868

= 119886119879minus2

120574119879minus2

+ 120574119879minus1

119864 (119876119879minus2

1198771015840

119879minus2) (Σ1

119879minus2)minus1

119868

119886119879minus2

= 119886119879minus2

120574119879minus2

(B1)


119879minus2

1015840Σ1




and 119897119879minus2

119909119879minus2

in 119906lowast

119879minus2

1015840Σ1



119879minus2is

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1 119868

119886119879minus2

=1

119886119879minus2

(B2)


is

1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

= 1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

minus 1205821

119879minus2120574119879minus2

1198681015840(Σ1

119879minus2)minus1

[120583119879minus2


119868]

= 1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

(B3)


is

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]1015840

sdot (Σ1

119879minus2)minus1

sdot [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

= [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]


1205742

119879minus2

(B4)


(1205821

119879minus2)2

4[120583119879minus2


119868]1015840

(Σ1

119879minus2)minus1

[120583119879minus2


119868]

=(1205821

119879minus2)2

4120572119879minus2

(B5)




119879minus2


is

1205821

119879minus2

2

1205831015840

119879minus2(Σ1

119879minus2)minus1

119868

119886119879minus2

=1205821

119879minus2119902119879minus2

2 (B6)



119909119879minus2

is 120574119879minus2


is

1205821

119879minus2

2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

(120583119879minus2


119868)

+1205821

119879minus2

21205831015840

119879minus2(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

(B7)


is

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

= [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)] minus 119886119879minus2

1205742

119879minus2

(B8)


(1205821

119879minus2

2)

2

1205831015840

119879minus2(Σ1

119879minus2)minus1

(120583119879minus2


119868) = (1205821

119879minus2

2)

2

120572119879minus2

(B9)


119879minus2(119909119879minus2

119897119879minus2


119879minus2is

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1 119868

119886119879minus2

=1

119886119879minus2

(B10)



119902119879minus2


119909119879minus2



is

[Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] + 119886

119879minus21205742

119879minus2

minus [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]

= 1205732119879minus2

(B11)


is

minus 1205821

119879minus21205831015840

119879minus2(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)]

= 1205731119879minus2

(B12)


minus1205822

119879minus1

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(B13)

Thus

119881lowast

119879minus2(119909119879minus2

119897119879minus2

) =1

119886119879minus2

1199092


119879minus2119902119879minus2

119909119879minus2


119909119879minus2

119897119879minus2

+ 1205731119879minus2

119897119879minus2

+ 1205732119879minus2

1198972

119879minus2minus

1205822

119879minus1

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(B14)


1198681015840

119886119905minus1

(Σ1

119905minus1)minus1

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

=119889119905minus1

119886119905minus1

+ 120574119905

1198681015840(Σ1

119905minus1)minus1

119864 (119876119905minus1

119877119905minus1

)

119886119905minus1


= 120574119905minus1

+ 120574119905

1198681015840(Σ1

119905minus1)minus1

119864 (119876119905minus1

119877119905minus1

)

119886119905minus1


= 120574119905minus1


= 0

1198681015840

119886119905minus1

(Σ1

119905minus1)minus1

[120583119905minus1


119868] =119887119905minus1

119886119905minus1


= 0

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

119868

= 119886119905minus1

120574119905minus1

+ 120574119905

119864 (119876119905minus1

1198771015840

119905minus1) (Σ1

119905minus1)minus1

119868

119886119905minus1

= 119886119905minus1

120574119905minus1

(C1)


119905minus1

1015840Σ1




and 119897119905minus1

119909119905minus1


is

1198681015840

119886119905minus1

(Σ1

119905minus1)minus1 119868

119886119905minus1

=1

119886119905minus1

(C2)


is

1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

= 1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

minus 1205821

119905minus1120574119905minus1

1198681015840(Σ1

119905minus1)minus1

[120583119905minus1


119868]


= 1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

(C3)


is

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]1015840

sdot (Σ1

119905minus1)minus1

sdot [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

= [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] minus 119886119905minus1

1205742

119905minus1

(C4)


(1205821

119905minus1)2

4[120583119905minus1


119868]1015840

(Σ1

119905minus1)minus1

[120583119905minus1


119868] =(1205821

119905minus1)2

4120572119905minus1

(C5)




119905minus1is


is

1205821

119905minus1

2

1205831015840

119905minus1(Σ1

119905minus1)minus1

119868

119886119905minus1

=1205821

119905minus1119902119905minus1

2 (C6)


119909119905minus1

is 120574119905minus1


is

1205821

119905minus1

2[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1



119868) +1205821

119905minus1

21205831015840

119905minus1(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

(C7)


is

[Σ0

119905minus1+ 120574119905119864(119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

= [Σ0

119905minus1+ 120574119905119864(119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] minus 119886119905minus1

1205742

119905minus1

(C8)


(1205821

119905minus1

2)

2

120583119905minus1

(Σ1

119905minus1)minus1

(120583119905minus1


119868) = (1205821

119905minus1

2)

2

120572119905minus1

(C9)


119905minus1(119909119905minus1

119897119905minus1


119905minus1is 1119886



1

119905minus1119902119905minus1


119905minus1119909119905minus1



is

[Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] + 119886

119905minus11205742

119905minus1

minus [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]

= 1205732119905minus1

(C10)


is

minus 1205821

119905minus11205831015840

119905minus1(Σ1

119905minus1)minus1

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

+ (120582119905minus1

+ 1205731119905) 119864 (119876

119905minus1) = 1205731119905minus1

(C11)


(1205821

119894)2120572119894

Thus

119881lowast

119905minus1(119909119905minus1

119897119905minus1

) =1

119886119905minus1

1199092


119905minus1119902119905minus1

119909119905minus1


119909119905minus1

119897119905minus1

+ 1205731119905minus1

119897119905minus1

+ 1205732119905minus1

1198972

119905minus1minus

1

4

119879minus1

sum119894=119905minus1

(1205821

119894)2

120572119894

(C12)

D Proof of (62)


119879minus1into 119906

lowast

119879minus1

1015840Σ119879minus1

119906lowast


119906lowast

119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1= (1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840

119879minus1)


)minus1

(120593119879minus1

119897119879minus1

+120582119879minus1

2120598119879minus1

)

= 1205931015840


)minus1

120593119879minus1

1198972

119879minus1+ 120582119879minus1

1205981015840

119879minus1


)minus1

120593119879minus1

119897119879minus1

+(120582119879minus1

)2

4119872119879minus1

2 (1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840



119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1

(D1)

Thus

119881lowast

119879minus1(119909119879minus1

119897119879minus1


119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1+ Var (119876

119879minus1) 1198972

119879minus1


119904119879minus1

119909119879minus1

+ 120582119879minus1

119864 (119876119879minus1

) 119897119879minus1

= [Var (119876119879minus1

) minus 1205931015840


)minus1

120593119879minus1

] 1198972

119879minus1

+ 120582119879minus1

[119864 (119876119879minus1

) minus 1205981015840


)minus1

120593119879minus1

]



119904119879minus1

119909119879minus1


)2

4119872119879minus1

= 1205732119879minus1

1198972

119879minus1+ 1205731119879minus1

119897119879minus1


119904119879minus1

119909119879minus1


)2

4119872119879minus1

(D2)



lowast


119906lowast

119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2= (1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840

119879minus2)


)minus1

(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= 1205931015840


)minus1

120593119879minus2

1198972

119879minus2+ 1205821

119879minus21205981015840

119879minus2


)minus1

120593119879minus2

119897119879minus2

+(1205821

119879minus2)2

4119872119879minus2

2 1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840



119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2

(E1)Thus

119881lowast

119879minus2(119909119879minus2

119897119879minus2


119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2

+ [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

= 1205732119879minus2

1198972

119879minus2+ 1205731119879minus2

119897119879minus2

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

minus(1205821

119879minus2)2

4119872119879minus2

(E2)where1205731119879minus2

= (120582119879minus2

+ 1205731119879minus1

) 119864 (119876119879minus2

) minus 1205821

119879minus21205981015840


)minus1

120593119879minus2

1205732119879minus2

= Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2) minus 1205931015840


)minus1

120593119879minus2

(E3)


lowast

119905minus1that

119906lowast

119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1= (1205931015840

119879minus1119897119905minus1

+1205821

119905minus1

21205981015840

119879minus1)


)minus1

(120593119905minus1

119897119905minus1

+1205821

119905minus1

2120598119905minus1

)

= 1205931015840


)minus1

120593119905minus1

1198972

119905minus1+ 1205821

119905minus11205981015840

119879minus1


)minus1

120593119905minus1

119897119905minus1

+(1205821

119905minus1)2

4119872119905minus1

2 (1205931015840

119879minus1119897119905minus1

+1205821

119905minus1

21205981015840



119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1

(F1)

Thus

119881lowast

119905minus1(119909119905minus1

119897119905minus1


119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1

+ [Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] 1198972

119905minus1

+ [120582119905minus1

119864 (119876119905minus1

) + 1205731119905119864 (119876119905minus1

)] 119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

= 1205732119905minus1

1198972

119905minus1+ 1205731119905minus1

119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905minus1

(1205821

119894)2

4119872119894

(F2)

where

1205731119905minus1

= (120582119905minus1

+ 1205731119905) 119864 (119876

119905minus1) minus 1205821

119905minus11205981015840


)minus1

120593119905minus1

1205732119905minus1

= Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1) minus 1205931015840


)minus1

120593119905minus1

(F3)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













my119905

0 (

09031199090+ 1194119897

0

minus00661199090+ 0107119897

0

01631199090minus 0577119897

0

) (

10181199090+ 0449119897

0

minus00601199090+ 0150119897

0

00421199090minus 0599119897

0

)

1 (

10181199091minus 0372119897

1

minus00601199091+ 0198119897

1

00421199091minus 0633119897

1

) (

10181199091+ 0449119897

1

minus00601199091+ 0150119897

1

00421199091minus 0599119897

1

)




my119905

0 (

04441198970+ 7889

01501198970minus 0062

minus05991198970minus 3572

) (

04441198970+ 3867

01501198970minus 0031

minus05991198970minus 1751

)

1 (

04441198971+ 3867

01501198971minus 0031

minus05991198971minus 1751

) (

04441198971+ 3867

01501198971minus 0031

minus05991198971minus 1751

)




my119905

0 (

04441198970

01501198970

minus05991198970

) (

04441198970

01501198970

minus05991198970

)

1 (

04441198971

01501198971

minus05991198971

) (

04441198971

01501198971

minus05991198971

)


119905= 120583119905minus 119904119905119868 =

(0122 0206 0188)1015840 and120593

119905= Σ0

119905= (00006 00149 00050)





6 Conclusion


Appendices

A Proof of (20)


119879minus1into 119906

lowast

119879minus1

1015840Σ1


119879minus1gives

119906lowast

119879minus1

1015840

Σ1


119879minus1=

119868

119886119879minus1

119909119879minus1

+ [Σ0


119868] 119897119879minus1

+120582119879minus1

2[120583119879minus1


119868]

1015840

(Σ1

119879minus1)minus1

times 119868

119886119879minus1

119909119879minus1

+ [Σ0


119868]


+120582119879minus1

2[120583119879minus1


119868]

=1

119886119879minus1

1199092

119879minus1+ [Σ0


119868]1015840

times (Σ1

119879minus1)minus1

[Σ0


119868] 1198972

119879minus1

+ 120582119879minus1

[Σ0


119868]1015840

times (Σ1

119879minus1)minus1

[120583119879minus1


119868] 119897119879minus1

+(120582119879minus1

)2

4[120583119879minus1


119868]1015840

times (Σ1

119879minus1)minus1

[120583119879minus1


119868]

=1

119886119879minus1

1199092

119879minus1+ [(Σ

0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0

119879minus1


1205742

119879minus1] 1198972

119879minus1

+ [(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1


120574119879minus1

]


119897119879minus1

+(120582119879minus1

)2

4120572119879minus1

(A1)



0

119879minus1)1015840

119897119879minus1

+

(120582119879minus1

2)1205831015840



2 [(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840


119879minus1

=2

119886119879minus1

[(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1

119868119909119879minus1

+ 2 [(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1


times [Σ0


119868] 119897119879minus1

+ 120582119879minus1

[(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1



119868]

= 2120574119879minus1

119897119879minus1

119909119879minus1

+ 119902119879minus1

120582119879minus1

119909119879minus1

+ 2 [(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0


119889119879minus1

] 1198972

119879minus1

+ 2120582119879minus1

[(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1


120574119879minus1

] 119897119879minus1

+(120582119879minus1

)2

2120572119879minus1

(A2)

Thus119881lowast

119879minus1(119909119879minus1

119897119879minus1

)

=1

119886119879minus1

1199092


119902119879minus1

119909119879minus1


119909119879minus1

119897119879minus1

+ 120582119879minus1

[119864 (119876119879minus1

) minus (Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1

+119887119879minus1

120574119879minus1

] 119897119879minus1

+ [Var (119876119879minus1

) minus (Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0

119879minus1

+120574119879minus1

119889119879minus1

] 1198972

119879minus1minus

(120582119879minus1

)2

4120572119879minus1

=1

119886119879minus1

1199092


119902119879minus1

119909119879minus1


119909119879minus1

119897119879minus1

+ 1205731119879minus1

119897119879minus1

+ 1205732119879minus1

1198972

119879minus1minus

(120582119879minus1

)2

4120572119879minus1

(A3)


1198681015840

119886119879minus2

(Σ1

119879minus2)minus1

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

=119889119879minus2

119886119879minus2

+ 120574119879minus1

1198681015840(Σ1

119879minus2)minus1

119864 (119876119879minus2

119877119879minus2

)

119886119879minus2


= 120574119879minus2

+ 120574119879minus1

1198681015840(Σ1

119879minus2)minus1

119864 (119876119879minus2

119877119879minus2

)

119886119879minus2


= 120574119879minus2


= 0

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1

[120583119879minus2


119868] =119887119879minus2

119886119879minus2


= 0

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

119868

= 119886119879minus2

120574119879minus2

+ 120574119879minus1

119864 (119876119879minus2

1198771015840

119879minus2) (Σ1

119879minus2)minus1

119868

119886119879minus2

= 119886119879minus2

120574119879minus2

(B1)


119879minus2

1015840Σ1




and 119897119879minus2

119909119879minus2

in 119906lowast

119879minus2

1015840Σ1



119879minus2is

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1 119868

119886119879minus2

=1

119886119879minus2

(B2)


is

1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

= 1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

minus 1205821

119879minus2120574119879minus2

1198681015840(Σ1

119879minus2)minus1

[120583119879minus2


119868]

= 1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

(B3)


is

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]1015840

sdot (Σ1

119879minus2)minus1

sdot [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

= [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]


1205742

119879minus2

(B4)


(1205821

119879minus2)2

4[120583119879minus2


119868]1015840

(Σ1

119879minus2)minus1

[120583119879minus2


119868]

=(1205821

119879minus2)2

4120572119879minus2

(B5)




119879minus2


is

1205821

119879minus2

2

1205831015840

119879minus2(Σ1

119879minus2)minus1

119868

119886119879minus2

=1205821

119879minus2119902119879minus2

2 (B6)



119909119879minus2

is 120574119879minus2


is

1205821

119879minus2

2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

(120583119879minus2


119868)

+1205821

119879minus2

21205831015840

119879minus2(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

(B7)


is

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

= [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)] minus 119886119879minus2

1205742

119879minus2

(B8)


(1205821

119879minus2

2)

2

1205831015840

119879minus2(Σ1

119879minus2)minus1

(120583119879minus2


119868) = (1205821

119879minus2

2)

2

120572119879minus2

(B9)


119879minus2(119909119879minus2

119897119879minus2


119879minus2is

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1 119868

119886119879minus2

=1

119886119879minus2

(B10)



119902119879minus2


119909119879minus2



is

[Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] + 119886

119879minus21205742

119879minus2

minus [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]

= 1205732119879minus2

(B11)


is

minus 1205821

119879minus21205831015840

119879minus2(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)]

= 1205731119879minus2

(B12)


minus1205822

119879minus1

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(B13)

Thus

119881lowast

119879minus2(119909119879minus2

119897119879minus2

) =1

119886119879minus2

1199092


119879minus2119902119879minus2

119909119879minus2


119909119879minus2

119897119879minus2

+ 1205731119879minus2

119897119879minus2

+ 1205732119879minus2

1198972

119879minus2minus

1205822

119879minus1

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(B14)


1198681015840

119886119905minus1

(Σ1

119905minus1)minus1

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

=119889119905minus1

119886119905minus1

+ 120574119905

1198681015840(Σ1

119905minus1)minus1

119864 (119876119905minus1

119877119905minus1

)

119886119905minus1


= 120574119905minus1

+ 120574119905

1198681015840(Σ1

119905minus1)minus1

119864 (119876119905minus1

119877119905minus1

)

119886119905minus1


= 120574119905minus1


= 0

1198681015840

119886119905minus1

(Σ1

119905minus1)minus1

[120583119905minus1


119868] =119887119905minus1

119886119905minus1


= 0

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

119868

= 119886119905minus1

120574119905minus1

+ 120574119905

119864 (119876119905minus1

1198771015840

119905minus1) (Σ1

119905minus1)minus1

119868

119886119905minus1

= 119886119905minus1

120574119905minus1

(C1)


119905minus1

1015840Σ1




and 119897119905minus1

119909119905minus1


is

1198681015840

119886119905minus1

(Σ1

119905minus1)minus1 119868

119886119905minus1

=1

119886119905minus1

(C2)


is

1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

= 1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

minus 1205821

119905minus1120574119905minus1

1198681015840(Σ1

119905minus1)minus1

[120583119905minus1


119868]


= 1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

(C3)


is

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]1015840

sdot (Σ1

119905minus1)minus1

sdot [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

= [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] minus 119886119905minus1

1205742

119905minus1

(C4)


(1205821

119905minus1)2

4[120583119905minus1


119868]1015840

(Σ1

119905minus1)minus1

[120583119905minus1


119868] =(1205821

119905minus1)2

4120572119905minus1

(C5)




119905minus1is


is

1205821

119905minus1

2

1205831015840

119905minus1(Σ1

119905minus1)minus1

119868

119886119905minus1

=1205821

119905minus1119902119905minus1

2 (C6)


119909119905minus1

is 120574119905minus1


is

1205821

119905minus1

2[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1



119868) +1205821

119905minus1

21205831015840

119905minus1(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

(C7)


is

[Σ0

119905minus1+ 120574119905119864(119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

= [Σ0

119905minus1+ 120574119905119864(119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] minus 119886119905minus1

1205742

119905minus1

(C8)


(1205821

119905minus1

2)

2

120583119905minus1

(Σ1

119905minus1)minus1

(120583119905minus1


119868) = (1205821

119905minus1

2)

2

120572119905minus1

(C9)


119905minus1(119909119905minus1

119897119905minus1


119905minus1is 1119886



1

119905minus1119902119905minus1


119905minus1119909119905minus1



is

[Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] + 119886

119905minus11205742

119905minus1

minus [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]

= 1205732119905minus1

(C10)


is

minus 1205821

119905minus11205831015840

119905minus1(Σ1

119905minus1)minus1

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

+ (120582119905minus1

+ 1205731119905) 119864 (119876

119905minus1) = 1205731119905minus1

(C11)


(1205821

119894)2120572119894

Thus

119881lowast

119905minus1(119909119905minus1

119897119905minus1

) =1

119886119905minus1

1199092


119905minus1119902119905minus1

119909119905minus1


119909119905minus1

119897119905minus1

+ 1205731119905minus1

119897119905minus1

+ 1205732119905minus1

1198972

119905minus1minus

1

4

119879minus1

sum119894=119905minus1

(1205821

119894)2

120572119894

(C12)

D Proof of (62)


119879minus1into 119906

lowast

119879minus1

1015840Σ119879minus1

119906lowast


119906lowast

119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1= (1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840

119879minus1)


)minus1

(120593119879minus1

119897119879minus1

+120582119879minus1

2120598119879minus1

)

= 1205931015840


)minus1

120593119879minus1

1198972

119879minus1+ 120582119879minus1

1205981015840

119879minus1


)minus1

120593119879minus1

119897119879minus1

+(120582119879minus1

)2

4119872119879minus1

2 (1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840



119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1

(D1)

Thus

119881lowast

119879minus1(119909119879minus1

119897119879minus1


119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1+ Var (119876

119879minus1) 1198972

119879minus1


119904119879minus1

119909119879minus1

+ 120582119879minus1

119864 (119876119879minus1

) 119897119879minus1

= [Var (119876119879minus1

) minus 1205931015840


)minus1

120593119879minus1

] 1198972

119879minus1

+ 120582119879minus1

[119864 (119876119879minus1

) minus 1205981015840


)minus1

120593119879minus1

]



119904119879minus1

119909119879minus1


)2

4119872119879minus1

= 1205732119879minus1

1198972

119879minus1+ 1205731119879minus1

119897119879minus1


119904119879minus1

119909119879minus1


)2

4119872119879minus1

(D2)



lowast


119906lowast

119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2= (1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840

119879minus2)


)minus1

(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= 1205931015840


)minus1

120593119879minus2

1198972

119879minus2+ 1205821

119879minus21205981015840

119879minus2


)minus1

120593119879minus2

119897119879minus2

+(1205821

119879minus2)2

4119872119879minus2

2 1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840



119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2

(E1)Thus

119881lowast

119879minus2(119909119879minus2

119897119879minus2


119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2

+ [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

= 1205732119879minus2

1198972

119879minus2+ 1205731119879minus2

119897119879minus2

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

minus(1205821

119879minus2)2

4119872119879minus2

(E2)where1205731119879minus2

= (120582119879minus2

+ 1205731119879minus1

) 119864 (119876119879minus2

) minus 1205821

119879minus21205981015840


)minus1

120593119879minus2

1205732119879minus2

= Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2) minus 1205931015840


)minus1

120593119879minus2

(E3)


lowast

119905minus1that

119906lowast

119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1= (1205931015840

119879minus1119897119905minus1

+1205821

119905minus1

21205981015840

119879minus1)


)minus1

(120593119905minus1

119897119905minus1

+1205821

119905minus1

2120598119905minus1

)

= 1205931015840


)minus1

120593119905minus1

1198972

119905minus1+ 1205821

119905minus11205981015840

119879minus1


)minus1

120593119905minus1

119897119905minus1

+(1205821

119905minus1)2

4119872119905minus1

2 (1205931015840

119879minus1119897119905minus1

+1205821

119905minus1

21205981015840



119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1

(F1)

Thus

119881lowast

119905minus1(119909119905minus1

119897119905minus1


119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1

+ [Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] 1198972

119905minus1

+ [120582119905minus1

119864 (119876119905minus1

) + 1205731119905119864 (119876119905minus1

)] 119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

= 1205732119905minus1

1198972

119905minus1+ 1205731119905minus1

119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905minus1

(1205821

119894)2

4119872119894

(F2)

where

1205731119905minus1

= (120582119905minus1

+ 1205731119905) 119864 (119876

119905minus1) minus 1205821

119905minus11205981015840


)minus1

120593119905minus1

1205732119905minus1

= Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1) minus 1205931015840


)minus1

120593119905minus1

(F3)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










times [Σ0


119868] 119897119879minus1

+ 120582119879minus1

[(Σ0

119879minus1)1015840

119897119879minus1

+120582119879minus1

21205831015840

119879minus1] (Σ1

119879minus1)minus1



119868]

= 2120574119879minus1

119897119879minus1

119909119879minus1

+ 119902119879minus1

120582119879minus1

119909119879minus1

+ 2 [(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0


119889119879minus1

] 1198972

119879minus1

+ 2120582119879minus1

[(Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1


120574119879minus1

] 119897119879minus1

+(120582119879minus1

)2

2120572119879minus1

(A2)

Thus119881lowast

119879minus1(119909119879minus1

119897119879minus1

)

=1

119886119879minus1

1199092


119902119879minus1

119909119879minus1


119909119879minus1

119897119879minus1

+ 120582119879minus1

[119864 (119876119879minus1

) minus (Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

120583119879minus1

+119887119879minus1

120574119879minus1

] 119897119879minus1

+ [Var (119876119879minus1

) minus (Σ0

119879minus1)1015840

(Σ1

119879minus1)minus1

Σ0

119879minus1

+120574119879minus1

119889119879minus1

] 1198972

119879minus1minus

(120582119879minus1

)2

4120572119879minus1

=1

119886119879minus1

1199092


119902119879minus1

119909119879minus1


119909119879minus1

119897119879minus1

+ 1205731119879minus1

119897119879minus1

+ 1205732119879minus1

1198972

119879minus1minus

(120582119879minus1

)2

4120572119879minus1

(A3)


1198681015840

119886119879minus2

(Σ1

119879minus2)minus1

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

=119889119879minus2

119886119879minus2

+ 120574119879minus1

1198681015840(Σ1

119879minus2)minus1

119864 (119876119879minus2

119877119879minus2

)

119886119879minus2


= 120574119879minus2

+ 120574119879minus1

1198681015840(Σ1

119879minus2)minus1

119864 (119876119879minus2

119877119879minus2

)

119886119879minus2


= 120574119879minus2


= 0

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1

[120583119879minus2


119868] =119887119879minus2

119886119879minus2


= 0

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

119868

= 119886119879minus2

120574119879minus2

+ 120574119879minus1

119864 (119876119879minus2

1198771015840

119879minus2) (Σ1

119879minus2)minus1

119868

119886119879minus2

= 119886119879minus2

120574119879minus2

(B1)


119879minus2

1015840Σ1




and 119897119879minus2

119909119879minus2

in 119906lowast

119879minus2

1015840Σ1



119879minus2is

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1 119868

119886119879minus2

=1

119886119879minus2

(B2)


is

1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

= 1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

minus 1205821

119879minus2120574119879minus2

1198681015840(Σ1

119879minus2)minus1

[120583119879minus2


119868]

= 1205821

119879minus2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[120583119879minus2


119868]

(B3)


is

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]1015840

sdot (Σ1

119879minus2)minus1

sdot [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

= [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]


1205742

119879minus2

(B4)


(1205821

119879minus2)2

4[120583119879minus2


119868]1015840

(Σ1

119879minus2)minus1

[120583119879minus2


119868]

=(1205821

119879minus2)2

4120572119879minus2

(B5)




119879minus2


is

1205821

119879minus2

2

1205831015840

119879minus2(Σ1

119879minus2)minus1

119868

119886119879minus2

=1205821

119879minus2119902119879minus2

2 (B6)



119909119879minus2

is 120574119879minus2


is

1205821

119879minus2

2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

(120583119879minus2


119868)

+1205821

119879minus2

21205831015840

119879minus2(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

(B7)


is

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

= [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)] minus 119886119879minus2

1205742

119879minus2

(B8)


(1205821

119879minus2

2)

2

1205831015840

119879minus2(Σ1

119879minus2)minus1

(120583119879minus2


119868) = (1205821

119879minus2

2)

2

120572119879minus2

(B9)


119879minus2(119909119879minus2

119897119879minus2


119879minus2is

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1 119868

119886119879minus2

=1

119886119879minus2

(B10)



119902119879minus2


119909119879minus2



is

[Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] + 119886

119879minus21205742

119879minus2

minus [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]

= 1205732119879minus2

(B11)


is

minus 1205821

119879minus21205831015840

119879minus2(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)]

= 1205731119879minus2

(B12)


minus1205822

119879minus1

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(B13)

Thus

119881lowast

119879minus2(119909119879minus2

119897119879minus2

) =1

119886119879minus2

1199092


119879minus2119902119879minus2

119909119879minus2


119909119879minus2

119897119879minus2

+ 1205731119879minus2

119897119879minus2

+ 1205732119879minus2

1198972

119879minus2minus

1205822

119879minus1

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(B14)


1198681015840

119886119905minus1

(Σ1

119905minus1)minus1

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

=119889119905minus1

119886119905minus1

+ 120574119905

1198681015840(Σ1

119905minus1)minus1

119864 (119876119905minus1

119877119905minus1

)

119886119905minus1


= 120574119905minus1

+ 120574119905

1198681015840(Σ1

119905minus1)minus1

119864 (119876119905minus1

119877119905minus1

)

119886119905minus1


= 120574119905minus1


= 0

1198681015840

119886119905minus1

(Σ1

119905minus1)minus1

[120583119905minus1


119868] =119887119905minus1

119886119905minus1


= 0

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

119868

= 119886119905minus1

120574119905minus1

+ 120574119905

119864 (119876119905minus1

1198771015840

119905minus1) (Σ1

119905minus1)minus1

119868

119886119905minus1

= 119886119905minus1

120574119905minus1

(C1)


119905minus1

1015840Σ1




and 119897119905minus1

119909119905minus1


is

1198681015840

119886119905minus1

(Σ1

119905minus1)minus1 119868

119886119905minus1

=1

119886119905minus1

(C2)


is

1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

= 1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

minus 1205821

119905minus1120574119905minus1

1198681015840(Σ1

119905minus1)minus1

[120583119905minus1


119868]


= 1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

(C3)


is

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]1015840

sdot (Σ1

119905minus1)minus1

sdot [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

= [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] minus 119886119905minus1

1205742

119905minus1

(C4)


(1205821

119905minus1)2

4[120583119905minus1


119868]1015840

(Σ1

119905minus1)minus1

[120583119905minus1


119868] =(1205821

119905minus1)2

4120572119905minus1

(C5)




119905minus1is


is

1205821

119905minus1

2

1205831015840

119905minus1(Σ1

119905minus1)minus1

119868

119886119905minus1

=1205821

119905minus1119902119905minus1

2 (C6)


119909119905minus1

is 120574119905minus1


is

1205821

119905minus1

2[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1



119868) +1205821

119905minus1

21205831015840

119905minus1(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

(C7)


is

[Σ0

119905minus1+ 120574119905119864(119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

= [Σ0

119905minus1+ 120574119905119864(119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] minus 119886119905minus1

1205742

119905minus1

(C8)


(1205821

119905minus1

2)

2

120583119905minus1

(Σ1

119905minus1)minus1

(120583119905minus1


119868) = (1205821

119905minus1

2)

2

120572119905minus1

(C9)


119905minus1(119909119905minus1

119897119905minus1


119905minus1is 1119886



1

119905minus1119902119905minus1


119905minus1119909119905minus1



is

[Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] + 119886

119905minus11205742

119905minus1

minus [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]

= 1205732119905minus1

(C10)


is

minus 1205821

119905minus11205831015840

119905minus1(Σ1

119905minus1)minus1

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

+ (120582119905minus1

+ 1205731119905) 119864 (119876

119905minus1) = 1205731119905minus1

(C11)


(1205821

119894)2120572119894

Thus

119881lowast

119905minus1(119909119905minus1

119897119905minus1

) =1

119886119905minus1

1199092


119905minus1119902119905minus1

119909119905minus1


119909119905minus1

119897119905minus1

+ 1205731119905minus1

119897119905minus1

+ 1205732119905minus1

1198972

119905minus1minus

1

4

119879minus1

sum119894=119905minus1

(1205821

119894)2

120572119894

(C12)

D Proof of (62)


119879minus1into 119906

lowast

119879minus1

1015840Σ119879minus1

119906lowast


119906lowast

119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1= (1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840

119879minus1)


)minus1

(120593119879minus1

119897119879minus1

+120582119879minus1

2120598119879minus1

)

= 1205931015840


)minus1

120593119879minus1

1198972

119879minus1+ 120582119879minus1

1205981015840

119879minus1


)minus1

120593119879minus1

119897119879minus1

+(120582119879minus1

)2

4119872119879minus1

2 (1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840



119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1

(D1)

Thus

119881lowast

119879minus1(119909119879minus1

119897119879minus1


119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1+ Var (119876

119879minus1) 1198972

119879minus1


119904119879minus1

119909119879minus1

+ 120582119879minus1

119864 (119876119879minus1

) 119897119879minus1

= [Var (119876119879minus1

) minus 1205931015840


)minus1

120593119879minus1

] 1198972

119879minus1

+ 120582119879minus1

[119864 (119876119879minus1

) minus 1205981015840


)minus1

120593119879minus1

]



119904119879minus1

119909119879minus1


)2

4119872119879minus1

= 1205732119879minus1

1198972

119879minus1+ 1205731119879minus1

119897119879minus1


119904119879minus1

119909119879minus1


)2

4119872119879minus1

(D2)



lowast


119906lowast

119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2= (1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840

119879minus2)


)minus1

(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= 1205931015840


)minus1

120593119879minus2

1198972

119879minus2+ 1205821

119879minus21205981015840

119879minus2


)minus1

120593119879minus2

119897119879minus2

+(1205821

119879minus2)2

4119872119879minus2

2 1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840



119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2

(E1)Thus

119881lowast

119879minus2(119909119879minus2

119897119879minus2


119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2

+ [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

= 1205732119879minus2

1198972

119879minus2+ 1205731119879minus2

119897119879minus2

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

minus(1205821

119879minus2)2

4119872119879minus2

(E2)where1205731119879minus2

= (120582119879minus2

+ 1205731119879minus1

) 119864 (119876119879minus2

) minus 1205821

119879minus21205981015840


)minus1

120593119879minus2

1205732119879minus2

= Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2) minus 1205931015840


)minus1

120593119879minus2

(E3)


lowast

119905minus1that

119906lowast

119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1= (1205931015840

119879minus1119897119905minus1

+1205821

119905minus1

21205981015840

119879minus1)


)minus1

(120593119905minus1

119897119905minus1

+1205821

119905minus1

2120598119905minus1

)

= 1205931015840


)minus1

120593119905minus1

1198972

119905minus1+ 1205821

119905minus11205981015840

119879minus1


)minus1

120593119905minus1

119897119905minus1

+(1205821

119905minus1)2

4119872119905minus1

2 (1205931015840

119879minus1119897119905minus1

+1205821

119905minus1

21205981015840



119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1

(F1)

Thus

119881lowast

119905minus1(119909119905minus1

119897119905minus1


119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1

+ [Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] 1198972

119905minus1

+ [120582119905minus1

119864 (119876119905minus1

) + 1205731119905119864 (119876119905minus1

)] 119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

= 1205732119905minus1

1198972

119905minus1+ 1205731119905minus1

119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905minus1

(1205821

119894)2

4119872119894

(F2)

where

1205731119905minus1

= (120582119905minus1

+ 1205731119905) 119864 (119876

119905minus1) minus 1205821

119905minus11205981015840


)minus1

120593119905minus1

1205732119905minus1

= Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1) minus 1205931015840


)minus1

120593119905minus1

(F3)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119909119879minus2

is 120574119879minus2


is

1205821

119879minus2

2[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

times (Σ1

119879minus2)minus1

(120583119879minus2


119868)

+1205821

119879minus2

21205831015840

119879minus2(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

(B7)


is

[Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

= [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)] minus 119886119879minus2

1205742

119879minus2

(B8)


(1205821

119879minus2

2)

2

1205831015840

119879minus2(Σ1

119879minus2)minus1

(120583119879minus2


119868) = (1205821

119879minus2

2)

2

120572119879minus2

(B9)


119879minus2(119909119879minus2

119897119879minus2


119879minus2is

1198681015840

119886119879minus2

(Σ1

119879minus2)minus1 119868

119886119879minus2

=1

119886119879minus2

(B10)



119902119879minus2


119909119879minus2



is

[Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] + 119886

119879minus21205742

119879minus2

minus [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]1015840

(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

)]

= 1205732119879minus2

(B11)


is

minus 1205821

119879minus21205831015840

119879minus2(Σ1

119879minus2)minus1

times [Σ0

119879minus2+ 120574119879minus1

119864 (119876119879minus2

119877119879minus2

) minus 120574119879minus2

119868]

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)]

= 1205731119879minus2

(B12)


minus1205822

119879minus1

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(B13)

Thus

119881lowast

119879minus2(119909119879minus2

119897119879minus2

) =1

119886119879minus2

1199092


119879minus2119902119879minus2

119909119879minus2


119909119879minus2

119897119879minus2

+ 1205731119879minus2

119897119879minus2

+ 1205732119879minus2

1198972

119879minus2minus

1205822

119879minus1

4120572119879minus1

minus(1205821

119879minus2)2

4120572119879minus2

(B14)


1198681015840

119886119905minus1

(Σ1

119905minus1)minus1

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

=119889119905minus1

119886119905minus1

+ 120574119905

1198681015840(Σ1

119905minus1)minus1

119864 (119876119905minus1

119877119905minus1

)

119886119905minus1


= 120574119905minus1

+ 120574119905

1198681015840(Σ1

119905minus1)minus1

119864 (119876119905minus1

119877119905minus1

)

119886119905minus1


= 120574119905minus1


= 0

1198681015840

119886119905minus1

(Σ1

119905minus1)minus1

[120583119905minus1


119868] =119887119905minus1

119886119905minus1


= 0

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

119868

= 119886119905minus1

120574119905minus1

+ 120574119905

119864 (119876119905minus1

1198771015840

119905minus1) (Σ1

119905minus1)minus1

119868

119886119905minus1

= 119886119905minus1

120574119905minus1

(C1)


119905minus1

1015840Σ1




and 119897119905minus1

119909119905minus1


is

1198681015840

119886119905minus1

(Σ1

119905minus1)minus1 119868

119886119905minus1

=1

119886119905minus1

(C2)


is

1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

= 1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

minus 1205821

119905minus1120574119905minus1

1198681015840(Σ1

119905minus1)minus1

[120583119905minus1


119868]


= 1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

(C3)


is

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]1015840

sdot (Σ1

119905minus1)minus1

sdot [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

= [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] minus 119886119905minus1

1205742

119905minus1

(C4)


(1205821

119905minus1)2

4[120583119905minus1


119868]1015840

(Σ1

119905minus1)minus1

[120583119905minus1


119868] =(1205821

119905minus1)2

4120572119905minus1

(C5)




119905minus1is


is

1205821

119905minus1

2

1205831015840

119905minus1(Σ1

119905minus1)minus1

119868

119886119905minus1

=1205821

119905minus1119902119905minus1

2 (C6)


119909119905minus1

is 120574119905minus1


is

1205821

119905minus1

2[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1



119868) +1205821

119905minus1

21205831015840

119905minus1(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

(C7)


is

[Σ0

119905minus1+ 120574119905119864(119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

= [Σ0

119905minus1+ 120574119905119864(119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] minus 119886119905minus1

1205742

119905minus1

(C8)


(1205821

119905minus1

2)

2

120583119905minus1

(Σ1

119905minus1)minus1

(120583119905minus1


119868) = (1205821

119905minus1

2)

2

120572119905minus1

(C9)


119905minus1(119909119905minus1

119897119905minus1


119905minus1is 1119886



1

119905minus1119902119905minus1


119905minus1119909119905minus1



is

[Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] + 119886

119905minus11205742

119905minus1

minus [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]

= 1205732119905minus1

(C10)


is

minus 1205821

119905minus11205831015840

119905minus1(Σ1

119905minus1)minus1

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

+ (120582119905minus1

+ 1205731119905) 119864 (119876

119905minus1) = 1205731119905minus1

(C11)


(1205821

119894)2120572119894

Thus

119881lowast

119905minus1(119909119905minus1

119897119905minus1

) =1

119886119905minus1

1199092


119905minus1119902119905minus1

119909119905minus1


119909119905minus1

119897119905minus1

+ 1205731119905minus1

119897119905minus1

+ 1205732119905minus1

1198972

119905minus1minus

1

4

119879minus1

sum119894=119905minus1

(1205821

119894)2

120572119894

(C12)

D Proof of (62)


119879minus1into 119906

lowast

119879minus1

1015840Σ119879minus1

119906lowast


119906lowast

119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1= (1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840

119879minus1)


)minus1

(120593119879minus1

119897119879minus1

+120582119879minus1

2120598119879minus1

)

= 1205931015840


)minus1

120593119879minus1

1198972

119879minus1+ 120582119879minus1

1205981015840

119879minus1


)minus1

120593119879minus1

119897119879minus1

+(120582119879minus1

)2

4119872119879minus1

2 (1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840



119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1

(D1)

Thus

119881lowast

119879minus1(119909119879minus1

119897119879minus1


119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1+ Var (119876

119879minus1) 1198972

119879minus1


119904119879minus1

119909119879minus1

+ 120582119879minus1

119864 (119876119879minus1

) 119897119879minus1

= [Var (119876119879minus1

) minus 1205931015840


)minus1

120593119879minus1

] 1198972

119879minus1

+ 120582119879minus1

[119864 (119876119879minus1

) minus 1205981015840


)minus1

120593119879minus1

]



119904119879minus1

119909119879minus1


)2

4119872119879minus1

= 1205732119879minus1

1198972

119879minus1+ 1205731119879minus1

119897119879minus1


119904119879minus1

119909119879minus1


)2

4119872119879minus1

(D2)



lowast


119906lowast

119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2= (1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840

119879minus2)


)minus1

(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= 1205931015840


)minus1

120593119879minus2

1198972

119879minus2+ 1205821

119879minus21205981015840

119879minus2


)minus1

120593119879minus2

119897119879minus2

+(1205821

119879minus2)2

4119872119879minus2

2 1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840



119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2

(E1)Thus

119881lowast

119879minus2(119909119879minus2

119897119879minus2


119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2

+ [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

= 1205732119879minus2

1198972

119879minus2+ 1205731119879minus2

119897119879minus2

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

minus(1205821

119879minus2)2

4119872119879minus2

(E2)where1205731119879minus2

= (120582119879minus2

+ 1205731119879minus1

) 119864 (119876119879minus2

) minus 1205821

119879minus21205981015840


)minus1

120593119879minus2

1205732119879minus2

= Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2) minus 1205931015840


)minus1

120593119879minus2

(E3)


lowast

119905minus1that

119906lowast

119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1= (1205931015840

119879minus1119897119905minus1

+1205821

119905minus1

21205981015840

119879minus1)


)minus1

(120593119905minus1

119897119905minus1

+1205821

119905minus1

2120598119905minus1

)

= 1205931015840


)minus1

120593119905minus1

1198972

119905minus1+ 1205821

119905minus11205981015840

119879minus1


)minus1

120593119905minus1

119897119905minus1

+(1205821

119905minus1)2

4119872119905minus1

2 (1205931015840

119879minus1119897119905minus1

+1205821

119905minus1

21205981015840



119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1

(F1)

Thus

119881lowast

119905minus1(119909119905minus1

119897119905minus1


119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1

+ [Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] 1198972

119905minus1

+ [120582119905minus1

119864 (119876119905minus1

) + 1205731119905119864 (119876119905minus1

)] 119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

= 1205732119905minus1

1198972

119905minus1+ 1205731119905minus1

119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905minus1

(1205821

119894)2

4119872119894

(F2)

where

1205731119905minus1

= (120582119905minus1

+ 1205731119905) 119864 (119876

119905minus1) minus 1205821

119905minus11205981015840


)minus1

120593119905minus1

1205732119905minus1

= Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1) minus 1205931015840


)minus1

120593119905minus1

(F3)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










= 1205821

119905minus1[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

times (Σ1

119905minus1)minus1

[120583119905minus1


119868]

(C3)


is

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]1015840

sdot (Σ1

119905minus1)minus1

sdot [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

= [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] minus 119886119905minus1

1205742

119905minus1

(C4)


(1205821

119905minus1)2

4[120583119905minus1


119868]1015840

(Σ1

119905minus1)minus1

[120583119905minus1


119868] =(1205821

119905minus1)2

4120572119905minus1

(C5)




119905minus1is


is

1205821

119905minus1

2

1205831015840

119905minus1(Σ1

119905minus1)minus1

119868

119886119905minus1

=1205821

119905minus1119902119905minus1

2 (C6)


119909119905minus1

is 120574119905minus1


is

1205821

119905minus1

2[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1



119868) +1205821

119905minus1

21205831015840

119905minus1(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

(C7)


is

[Σ0

119905minus1+ 120574119905119864(119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

= [Σ0

119905minus1+ 120574119905119864(119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)] minus 119886119905minus1

1205742

119905minus1

(C8)


(1205821

119905minus1

2)

2

120583119905minus1

(Σ1

119905minus1)minus1

(120583119905minus1


119868) = (1205821

119905minus1

2)

2

120572119905minus1

(C9)


119905minus1(119909119905minus1

119897119905minus1


119905minus1is 1119886



1

119905minus1119902119905minus1


119905minus1119909119905minus1



is

[Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] + 119886

119905minus11205742

119905minus1

minus [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]1015840

(Σ1

119905minus1)minus1

times [Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

)]

= 1205732119905minus1

(C10)


is

minus 1205821

119905minus11205831015840

119905minus1(Σ1

119905minus1)minus1

[Σ0

119905minus1+ 120574119905119864 (119876119905minus1

119877119905minus1

) minus 120574119905minus1

119868]

+ (120582119905minus1

+ 1205731119905) 119864 (119876

119905minus1) = 1205731119905minus1

(C11)


(1205821

119894)2120572119894

Thus

119881lowast

119905minus1(119909119905minus1

119897119905minus1

) =1

119886119905minus1

1199092


119905minus1119902119905minus1

119909119905minus1


119909119905minus1

119897119905minus1

+ 1205731119905minus1

119897119905minus1

+ 1205732119905minus1

1198972

119905minus1minus

1

4

119879minus1

sum119894=119905minus1

(1205821

119894)2

120572119894

(C12)

D Proof of (62)


119879minus1into 119906

lowast

119879minus1

1015840Σ119879minus1

119906lowast


119906lowast

119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1= (1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840

119879minus1)


)minus1

(120593119879minus1

119897119879minus1

+120582119879minus1

2120598119879minus1

)

= 1205931015840


)minus1

120593119879minus1

1198972

119879minus1+ 120582119879minus1

1205981015840

119879minus1


)minus1

120593119879minus1

119897119879minus1

+(120582119879minus1

)2

4119872119879minus1

2 (1205931015840

119879minus1119897119879minus1

+120582119879minus1

21205981015840



119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1

(D1)

Thus

119881lowast

119879minus1(119909119879minus1

119897119879minus1


119879minus1

1015840

Σ119879minus1

119906lowast

119879minus1+ Var (119876

119879minus1) 1198972

119879minus1


119904119879minus1

119909119879minus1

+ 120582119879minus1

119864 (119876119879minus1

) 119897119879minus1

= [Var (119876119879minus1

) minus 1205931015840


)minus1

120593119879minus1

] 1198972

119879minus1

+ 120582119879minus1

[119864 (119876119879minus1

) minus 1205981015840


)minus1

120593119879minus1

]



119904119879minus1

119909119879minus1


)2

4119872119879minus1

= 1205732119879minus1

1198972

119879minus1+ 1205731119879minus1

119897119879minus1


119904119879minus1

119909119879minus1


)2

4119872119879minus1

(D2)



lowast


119906lowast

119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2= (1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840

119879minus2)


)minus1

(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= 1205931015840


)minus1

120593119879minus2

1198972

119879minus2+ 1205821

119879minus21205981015840

119879minus2


)minus1

120593119879minus2

119897119879minus2

+(1205821

119879minus2)2

4119872119879minus2

2 1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840



119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2

(E1)Thus

119881lowast

119879minus2(119909119879minus2

119897119879minus2


119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2

+ [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

= 1205732119879minus2

1198972

119879minus2+ 1205731119879minus2

119897119879minus2

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

minus(1205821

119879minus2)2

4119872119879minus2

(E2)where1205731119879minus2

= (120582119879minus2

+ 1205731119879minus1

) 119864 (119876119879minus2

) minus 1205821

119879minus21205981015840


)minus1

120593119879minus2

1205732119879minus2

= Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2) minus 1205931015840


)minus1

120593119879minus2

(E3)


lowast

119905minus1that

119906lowast

119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1= (1205931015840

119879minus1119897119905minus1

+1205821

119905minus1

21205981015840

119879minus1)


)minus1

(120593119905minus1

119897119905minus1

+1205821

119905minus1

2120598119905minus1

)

= 1205931015840


)minus1

120593119905minus1

1198972

119905minus1+ 1205821

119905minus11205981015840

119879minus1


)minus1

120593119905minus1

119897119905minus1

+(1205821

119905minus1)2

4119872119905minus1

2 (1205931015840

119879minus1119897119905minus1

+1205821

119905minus1

21205981015840



119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1

(F1)

Thus

119881lowast

119905minus1(119909119905minus1

119897119905minus1


119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1

+ [Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] 1198972

119905minus1

+ [120582119905minus1

119864 (119876119905minus1

) + 1205731119905119864 (119876119905minus1

)] 119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

= 1205732119905minus1

1198972

119905minus1+ 1205731119905minus1

119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905minus1

(1205821

119894)2

4119872119894

(F2)

where

1205731119905minus1

= (120582119905minus1

+ 1205731119905) 119864 (119876

119905minus1) minus 1205821

119905minus11205981015840


)minus1

120593119905minus1

1205732119905minus1

= Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1) minus 1205931015840


)minus1

120593119905minus1

(F3)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











lowast


119906lowast

119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2= (1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840

119879minus2)


)minus1

(120593119879minus2

119897119879minus2

+1205821

119879minus2

2120598119879minus2

)

= 1205931015840


)minus1

120593119879minus2

1198972

119879minus2+ 1205821

119879minus21205981015840

119879minus2


)minus1

120593119879minus2

119897119879minus2

+(1205821

119879minus2)2

4119872119879minus2

2 1205931015840

119879minus2119897119879minus2

+1205821

119879minus2

21205981015840



119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2

(E1)Thus

119881lowast

119879minus2(119909119879minus2

119897119879minus2


119879minus2

1015840

Σ119879minus2

119906lowast

119879minus2

+ [Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2)] 1198972

119879minus2

+ [120582119879minus2

119864 (119876119879minus2

) + 1205731119879minus1

119864 (119876119879minus2

)] 119897119879minus1

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

= 1205732119879minus2

1198972

119879minus2+ 1205731119879minus2

119897119879minus2

minus 1205821

119879minus2119904119879minus2

119909119879minus2


)2

4119872119879minus1

minus(1205821

119879minus2)2

4119872119879minus2

(E2)where1205731119879minus2

= (120582119879minus2

+ 1205731119879minus1

) 119864 (119876119879minus2

) minus 1205821

119879minus21205981015840


)minus1

120593119879minus2

1205732119879minus2

= Var (119876119879minus2

) + 1205732119879minus1

119864 (1198762

119879minus2) minus 1205931015840


)minus1

120593119879minus2

(E3)


lowast

119905minus1that

119906lowast

119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1= (1205931015840

119879minus1119897119905minus1

+1205821

119905minus1

21205981015840

119879minus1)


)minus1

(120593119905minus1

119897119905minus1

+1205821

119905minus1

2120598119905minus1

)

= 1205931015840


)minus1

120593119905minus1

1198972

119905minus1+ 1205821

119905minus11205981015840

119879minus1


)minus1

120593119905minus1

119897119905minus1

+(1205821

119905minus1)2

4119872119905minus1

2 (1205931015840

119879minus1119897119905minus1

+1205821

119905minus1

21205981015840



119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1

(F1)

Thus

119881lowast

119905minus1(119909119905minus1

119897119905minus1


119905minus1

1015840

Σ119905minus1

119906lowast

119905minus1

+ [Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1)] 1198972

119905minus1

+ [120582119905minus1

119864 (119876119905minus1

) + 1205731119905119864 (119876119905minus1

)] 119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905

(1205821

119894)2

4119872119894

= 1205732119905minus1

1198972

119905minus1+ 1205731119905minus1

119897119905minus1

minus 1205821

119905minus1119904119905minus1

119909119905minus1

minus

119879minus1

sum119894=119905minus1

(1205821

119894)2

4119872119894

(F2)

where

1205731119905minus1

= (120582119905minus1

+ 1205731119905) 119864 (119876

119905minus1) minus 1205821

119905minus11205981015840


)minus1

120593119905minus1

1205732119905minus1

= Var (119876119905minus1

) + 1205732119905119864 (1198762

119905minus1) minus 1205931015840


)minus1

120593119905minus1

(F3)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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