research article tightness criterion and weak...
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Hindawi Publishing CorporationISRN Probability and StatisticsVolume 2013 Article ID 543723 12 pageshttpdxdoiorg1011552013543723
Research ArticleTightness Criterion and Weak Convergence forthe Generalized Empirical Process in 119863[0 1]
Maciej Ziemba
Department of Mathematics Lublin University of Technology ul Nadbystrzycka 38d20-618 Lublin Poland
Correspondence should be addressed to Maciej Ziemba maciekziembagmailcom
Received 27 June 2013 Accepted 23 August 2013
Academic Editors M Campanino S Lototsky H J Paarsch and L Sacerdote
Copyright copy 2013 Maciej Ziemba 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
We prove Shao and Yursquos tightness criterion for the generalized empirical process in the space 119863[0 1] with 1198691topology Covariance
inequalities are used in applying the criterion to particular types of the empirical processes We weaken the assumptions imposedon the covariance structure as well as the properties of the underlying sequence of rvrsquos under which presented processes convergeweakly
1 Introduction
Let 119883119899119899ge1
be a sequence of absolutely continuous identicallydistributed (id) random variables (rvrsquos) with an unknowndistribution function (df)119865 and probability density function(pdf) 119891 The empirical distribution function based on thefirst 119899 rvrsquos is defined by 119865
119899(119909) = 119899
minus1sum
119899
119895=1119868[119883
119895le 119909] It is
well known however that this estimate does not make useof the smoothness of 119865 that is the existence of the pdf 119891Therefore the kernel estimate
119865119899(119909) =
1
119899
119899
sum
119895=1
119870(
119909 minus 119883119895
ℎ119899
) (1)
has been proposed where the kernel function 119870 is a knowndf and ℎ
119899119899ge1
is a sequence of positive constants descendingat an appropriate rate Such estimator has been deeply studiedin the last two decades mainly by Cai and Roussas in [1ndash4]Li and Yang in [5] and others Asymptotic normality Berry-Essen bounds for smooth estimator 119865
119899(119909) are only examples
of their fruitful resultsRecently Li et al proposed in [6] the so-called recursive
kernel estimator of the df 119865 as follows
119865119899(119909) =
1
119899
119899
sum
119895=1
119870(
119909 minus 119883119895
ℎ119895
) (2)
The seemingly tiny modification they introduced to theformula of the typical kernel estimator has an importantadvantage Namely in the case of a large size of a sample119865119899(119909) can be easily updated with each new observation since
it is computable recursively by
119865119899(119909) =
119899 minus 1
119899119865119899minus1
(119909) +1
119899119870(
119909 minus 119883119899
ℎ119899
) (3)
where 1198650(119909) = 0 The authors discussed the asymptotic bias
and quadratic-mean convergence and established the point-wise asymptotic normality of 119865
119899(119909) under relevant assump-
tionsIn this paper however we will focus on the empirical
process built on an estimator 119865119899(119909) of the df 119865 rather than
119865119899(119909) itself Let us recall that the following process
120572119899(119909) = radic119899 [119865
119899(119909) minus 119864119865
119899(119909)] where 119909 isin R (4)
is called the empirical process built on an estimator 119865119899(119909)
Yu [7] studied the case when119865119899(119909) is a standard empirical
df and showed weak convergence of 120572119899(sdot) to the Gaussian
process assuming stationarity and association of the underly-ing rvrsquos Cai and Roussas [1] obtained a similar result in thecase when 119865
119899(119909) is the kernel estimator of the df 119865 built on
a stationary sequence of negatively associated rvrsquos
2 ISRN Probability and Statistics
In this paper we shall study the empirical process 120572119899(119909)
generated by the generalized kernel estimator of the df givenby the formula
119865119899(119909) =
1
119899
119899
sum
119895=1
119870(
119909 minus 119883119895
ℎ119899119895
) where (5)
A1 119883119895119895ge1
is a sequence of absolutely continuous id rvrsquostaking values in [0 1] and having twice differentiabledf 119865 with first and second derivative bounded
A2 119870 is a kernel function such that intR 119906119889119870(119906) = 0 andintR 119906
2119889119870(119906) lt infin with bounded derivative 119896
A3 ℎ119899119895119899ge1119895isin1119899
is a sequence of positive constantssubject to the following conditions lim
119895119899rarrinfinℎ119899119895
= 0lim
119895119899rarrinfin119899ℎ
4
119899119895= 0 (actually since 119895 le 119899 119895 rarr infin
under the limit suffices)
Explicitly we take a look onto the process
120572119899(119909) =
1
radic119899sdot
119899
sum
119895=1
[119870(
119909 minus 119883119895
ℎ119899119895
) minus 119864119870(
119909 minus 119883119895
ℎ119899119895
)] (6)
we shall call from now on the generalized empirical processLet us pay attention to the fact that in the case of
(i) ℎ119899119895
= ℎ119899for 119895 isin 1 119899 120572
119899(119909) is the empirical
process based on the kernel estimator of the df 119865(ii) ℎ
119895119895= ℎ
119895+1119895= ℎ
119895+2119895= sdot sdot sdot = ℎ
119895for 119895 isin N 120572
119899(119909)
is the empirical process based on the recursive kernelestimator of the df 119865
(iii) 119870(119905) = 119868[0infin]
(119905) 120572119899(119909) is the standard empirical
process (based on the empirical df)
It is well known that the crucial procedure in showingweak convergence for an empirical process is to verify tight-ness In [8] Shao and Yu gave the following criterion
exist1198621gt0exist
119901gt2exist
1199011gt1exist
0le1199031le1exist
1199012gt1minus1199031
forall119909119910isin[01]
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
le 1198621(1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
1199011
+ 119899minus119901221003816100381610038161003816119909 minus 119910
1003816100381610038161003816
1199031
)
(7)
under which the standard empirical process based on sta-tionary sequence of uniform [0 1] rvrsquos is tight It is statedthere that the proof of that fact is an easy standard procedureparallel to the one presented in [9] It is the main aim of thispaper to carry it in details but for the generalized empiricalprocess defined by (6) and without assuming stationarityNevertheless wewill always return to stationarity assumptionwhile establishing weak convergence
In order to obtain tightness one has to assume appro-priate covariance structure of the underlying rvrsquos that isthe covariance of a pair of rvrsquos 119883
119894and 119883
119895has to decline
at the right rate while 119894 and 119895 are growing apart In thispaper we lower the demanded rate of covariance decay usingthe covariance inequalities for associated (cf [10 11]) andmultivariate totally positive of order 2 (MTP
2) (cf [12]) rvrsquos
obtained in [13]
The paper is organized as follows In Section 2 we presentthe proof of the Shao and Yursquos tightness criterion formulatedfor our generalized empirical process Sections 3 and 4 aredevoted to application of the criterion to showing tightnessand thus weak convergence of the specific types of empiricalprocesses Section 5 concerns weak convergence of the recur-sive kernel-type process for iid rvrsquos
2 Tightness Criterion
We start with the key point of the paper
Theorem 1 Let 120572119899(119909)
119899ge1isin 119863[0 1] be the generalized empir-
ical process defined as in (6) One assume that A1 A2 A3 holdIf there exist constants 119862 gt 0 119901 gt 2 119901
1gt 1 0 le 119901
3le 1
1199012gt 1 minus 119901
3 such that for any 119909 119910 isin [0 1] and 119899 isin N the
following inequality holds
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
le 119862 sdot (1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
1199011
+ 119899minus11990122sdot1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
1199013
)
(8)
then the process 120572119899(119909)
119899ge1is tight in119863[0 1] with 119869
1topology
Proof The proof boils down to showing that under theassumptions made inTheorem 1 conditions ofTheorem 132in [9] hold Let us recall that in light of the above mentionedtheorem a process 120572
119899(119909)
119899ge1is tight in119863[0 1] if
lim119886rarrinfin
lim sup119899rarrinfin
119875 (1003817100381710038171003817120572119899
1003817100381710038171003817 ge 119886) = 0 (9)
lim120575rarr0
lim sup119899rarrinfin
119875 (1199081015840
1(120572
119899 120575) ge 120598) = 0 forall120598 gt 0 (10)
where sdot is the supremum norm that is1003817100381710038171003817120572119899
1003817100381710038171003817 = sup0le119909le1
1003816100381610038161003816120572119899(119909)
1003816100381610038161003816 (11)
and 1199081015840
1(120572
119899 120575) is the modulus of continuity of the function
120572119899isin 119863[0 1] that is
1199081015840
1(120572
119899 120575) = inf
119905119894
max1le119894le]
sup119909119910isin[119905119894minus1119905119894)
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 (12)
The infimum runs over all finite ldquo120575-sparserdquo decompositions[119905
119894minus1 119905
119894) 119894 isin 1 ] of the interval [0 1] In other words it
runs over all choices of increasingly ordered points 1199051198941le119894le]
such that min1le119894le](119905119894 minus 119905
119894minus1) gt 120575 where 119905
0= 0 119905] = 1
Let us first show that condition (9) is satisfied Accordingto the corollary following Theorem 132 in [9] it suffices toshow
lim119886rarrinfin
lim sup119899rarrinfin
119875 (1205721198991003816100381610038161003816120572119899
(119909)1003816100381610038161003816 ge 119886) = 0 forall119909 isin [0 1] (13)
that is
forall120578ge0
exist119886gt0
forall119899isinN119875 (120572
1198991003816100381610038161003816120572119899
(119909)1003816100381610038161003816 ge 119886) le 120578 forall119909 isin [0 1] (14)
Let us fix 119909 isin [0 1] and 120578 gt 0 We need to find 119886 = 119886(120578) gt 0such that
119875 (radic11989910038161003816100381610038161003816119865119899(119909) minus 119864119865
119899(119909)
10038161003816100381610038161003816ge 119886) le 120578 forall119899 isin N (15)
ISRN Probability and Statistics 3
where
119865119899(119909) =
1
119899
119899
sum
119895=1
119870(
119909 minus 119883119895
ℎ119899119895
) (16)
is the generalized kernel estimator of the df119865 ApplyingChe-byshevrsquos inequality we shall find 119886 satisfying
119886 geradic119899
120578sdot 119864
10038161003816100381610038161003816119865119899(119909) minus 119864119865
119899(119909)
10038161003816100381610038161003816forall119899 isin N (17)
Such 119886 exists if 119864|120572119899(119909)| lt infin for all 119899 ge 1 Implementing
119910 = 0 and 1199013= 0 in the assumption (8) we get for 119901 gt 2
1199011 119901
2gt 1
1198641003816100381610038161003816120572119899
(119909)1003816100381610038161003816
119901
le 119862 sdot(|119909|1199011+119899
minus11990122|119909|
1199013) le 119862 sdot(1+
1
11989911990122) le 2119862
(18)
Now applying Holderrsquos inequality we arrive at
1198641003816100381610038161003816120572119899
(119909)1003816100381610038161003816 le
4radic119864
1003816100381610038161003816120572119899(119909)
1003816100381610038161003816
4
(19)
which in light of the inequality (18) for 119901 = 4 is boundedfrom above by 4radic2119862 lt infin Therefore condition (9) holds
We shall now proceed to checking condition (10) Let usrecall that the modulus of continuity of the function 119909 isin
119862[0 1] 120575 gt 0 is given by the formula
1199081(119909 120575) = sup
0le119910le1minus120575
sup119910le119909le119910+120575
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 (20)
As it is shown in [9] (page 123) there is a relation between119908
1(sdot sdot) and 1199081015840
1(sdot sdot) in the spaces 119862[0 1] and119863[0 1] relatively
Namely for 119909 isin 119863[0 1] and 120575 lt 12
1199081(119909 2120575) ge 119908
1015840
1(119909 120575) (21)
Thus
119875 (1199081015840
1(120572
119899 120575) ge 120598) le 119875 (119908
1(120572
119899 2120575) ge 120598) (22)
Therefore condition (10) holds if we show
forall120598gt0
forall120578gt0
exist0le120575lt12
exist1198990
forall119899ge1198990
119875 (1199081(120572
119899 2120575) ge 120598) le 120578 (23)
With a view to obtaining the desired inequality we shall pro-ceed patiently in five steps
Step 1 We need a moment inequality for the rv 120572119899(119909)minus120572
119899(119910)
involving the distance between the points 119909 and 119910 Let usthen assume that for the constants 119862 119901 119901
1 119901
2 119901
3given in
Theorem 1 inequality (8) holds Fix 120598 gt 0 120578 gt 0 and definethe quantity 119903
119899= 120598radic119899 119899 isin N Next we fix 119909 119910 isin [0 1] and
take 119899 large enough so that 119903119899le |119909minus119910|Then 119899minus12
le |119909minus119910|120598minus1
and we have
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
le 119862 (1 + 120598minus1199012)1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
min11990111199012+1199013
(24)
Step 2 Let us now fix 119899 isin N 120575 isin (0 1) and consider the fol-lowing rvrsquos
120594119894= 120572
119899(119910 + 119894119903
119899) minus 120572
119899(119910 + (119894 minus 1) 119903
119899) (25)
for 119894 isin 1 2 119898119899 where119898
119899= 119898(119899 120575) is such that 119903
119899119898
119899le 120575
It is easy to see that for 119878119894= sum
119894
119896=1120594
119896
max1le119894le119898
119899
1003816100381610038161003816119878119894
1003816100381610038161003816 = max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119903
119899) minus 120572
119899(119910)
1003816100381610038161003816 (26)
Let us notice that for rvrsquos 120594119894119894ge1
the conditions of Theo-rem 102 in [9] are satisfied with 4120573 = 119901 119906
119897= 119903
119899forall
119894lt119897le119895and
2120572 = min1199011 119901
2+ 119901
3 Therefore we are equipped with the
following maximal inequality
119875( max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119903
119899) minus 120572
119899(119910)
1003816100381610038161003816 ge 120582)
le
119862119901min119901
11199012+1199013
120582119901(119898
119899119903119899)min119901
11199012+1199013
(27)
for all 120582 gt 0
Step 3 Let 119872 = sup119909isin[01]
119891(119909) where 119891 is the pdf of rvrsquos119883
119895119895ge1
For fixed 120598 120578 gt 0 let us take 120575 gt 0 such that
119862119901min119901
11199012+1199013
(119872120598)119901
sdot (2120575)min119901
11199012+1199013lt 120578 (28)
and define119898119899= lfloor120575119903
119899rfloor For sufficiently large 119899 isin N we have
119898119899ge 1 and then we get
119903119899119898
119899le 120575 lt (119898
119899+ 1) 119903
119899le 2119898
119899119903119899le 120575 (29)
Step 4 Our goal is to obtain an inequality which enables usto bound the supremum of the increment of the function 120572
119899
via the maximum of the increments of that function on somesubintervals To bemore precise wewill find the upper boundfor
sup119910le119909le119910+119898
119899119902
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 (30)
in terms of
max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572
119899(119910)
1003816100381610038161003816 (31)
where 119909 119910 isin [0 1] 119910 le 119909 le 119910 + 119902 119902 ge 120598119899 and119898119899is defined
as in Step 3Let us recall that 120572
119899(119909) = radic119899(119865
119899(119909) minus 119864119865
119899(119909)) where
119865119899(119909) = (1119899)sum
119899
119895=1119870((119909 minus 119883
119895)ℎ
119899119895) From the triangle ine-
quality we can see that
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816
leradic119899
119899
10038161003816100381610038161003816100381610038161003816100381610038161003816
119899
sum
119895=1
[119870(
119909 minus 119883119895
ℎ119899119895
) minus 119870(
119910 minus 119883119895
ℎ119899119895
)]
10038161003816100381610038161003816100381610038161003816100381610038161003816
+ radic119899 (10038161003816100381610038161003816119864119865
119899(119909) minus 119865 (119909)
10038161003816100381610038161003816+10038161003816100381610038161003816119864119865
119899(119910) minus 119865 (119910)
10038161003816100381610038161003816
+1003816100381610038161003816119865 (119909) minus 119865 (119910)
1003816100381610038161003816 )
(32)
4 ISRN Probability and Statistics
Since 119870 is a nondecreasing function and applying triangleinequality again we get
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816
le1003816100381610038161003816120572119899
(119910 + 119902) minus 120572119899(119910)
1003816100381610038161003816 +radic119899
10038161003816100381610038161003816119864119865
119899(119910 + 119902) minus 119864119865
119899(119910)
10038161003816100381610038161003816
+ radic119899 (10038161003816100381610038161003816119864119865
119899(119909) minus 119865 (119909)
10038161003816100381610038161003816+10038161003816100381610038161003816119864119865
119899(119910) minus 119865 (119910)
10038161003816100381610038161003816
+1003816100381610038161003816119865 (119909) minus 119865 (119910)
1003816100381610038161003816 )
(33)
Cai and Roussas in [1] showed that under assumptions madeon the df 119865 and the kernel function119870 we have
10038161003816100381610038161003816119864119865
119899(119909) minus 119865 (119909)
10038161003816100381610038161003816= 119874 (ℎ
2
119899) (34)
Similarly by Taylor expansion it is easy to see that
1003816100381610038161003816119865 (119909) minus 119865 (119910)1003816100381610038161003816 le
10038171003817100381710038171198911003817100381710038171003817 sdot
1003816100381610038161003816119909 minus 1199101003816100381610038161003816 (35)
Thus
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 le1003816100381610038161003816120572119899
(119910 + 119902) minus 120572119899(119910)
1003816100381610038161003816
+ 119862radic119899ℎ4
119899+119872radic119899119902
(36)
where 119872 = sup119909isin[01]
119891(119909) and 119862 = 1198621198911015840119870
is a positive con-stant dependent on the functions 119865 and119870
Let us recall that 119910 le 119909 le 119910+ 119902119898119899= lfloor120575119903
119899rfloor and 119902 ge 120598119899
Taking 119902 = (120598radic119899)(= 119903119899) we notice that 119898
119899119902 = lfloor120575119903
119899rfloor119903
119899le 120575
thus when 119899 rarr infin the interval [119910 119910 + 119898119899119902] coincides with
[119910 119910 + 120575] Approaching the aim of Step 4 let us observe that
sup119910le119909le119910+119898
119899119902
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 = max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572
119899(119910)
1003816100381610038161003816
+1003816100381610038161003816120572119899
(1199090) minus 120572
119899(119910 + 119894max119902)
1003816100381610038161003816
(37)
where 1199090isin [119910+119894
0119902 119910+(119894
0+1)119902] is a point at which the above
supremum is attained and
119894max =
1198940+ 1 if 1003816100381610038161003816120572119899
(119910 + [1198940+ 1] 119902) minus 120572
119899(119910)
1003816100381610038161003816
ge1003816100381610038161003816120572119899
(119910 + 1198940119902) minus 120572
119899(119910)
1003816100381610038161003816
1198940
otherwise(38)
Without the loss of generality we may assume that 119894max = 1198940+
1 which implies
sup119910le119909le119910+119898
119899119902
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 = max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572
119899(119910)
1003816100381610038161003816
+1003816100381610038161003816120572119899
(1199090) minus 120572
119899(119910 + 119894
0119902)1003816100381610038161003816
(39)
Now applying inequality (36) the definition of 119894max and thetriangle inequality we have
sup119910le119909le119910+119898
119899119902
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816
le max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572
119899(119910)
1003816100381610038161003816
+1003816100381610038161003816120572119899
(119910 + [1198940+ 1] 119902) minus 120572
119899(119910 + 119894
0119902)1003816100381610038161003816
+ 119872radic119899119902 + 119862radic119899ℎ4
119899
le max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572
119899(119910)
1003816100381610038161003816
+1003816100381610038161003816120572119899
(119910 + [1198940+ 1] 119902) minus 120572
119899(119910)
1003816100381610038161003816
+1003816100381610038161003816120572119899
(119910 + 1198940119902) minus 120572
119899(119910)
1003816100381610038161003816 + 119872radic119899119902 + 119862radic119899ℎ4
119899
le 3 max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572
119899(119910)
1003816100381610038161003816 + 119872radic119899119902 + 119862radic119899ℎ4
119899
(40)
If we plug in 119902 = 119903119899we arrive at
sup119910le119909le119910+119898
119899119903119899
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 le 3 max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119903
119899) minus 120572
119899(119910)
1003816100381610038161003816
+ 119872radic119899119903119899+ 119862radic119899ℎ4
119899
(41)
Step 5 Finally we are in a position to obtain inequality (23)that is
forall120598gt0forall
120578gt0exist
0le120575lt12exist
1198990
forall119899ge1198990
119875( sup0le119910le1minus2120575
sup119910le119909le119910+2120575
1003816100381610038161003816120572119899(119909)minus120572
119899(119910)
1003816100381610038161003816
ge 120598) le 120578
(42)
We now successivelymake use the inequalities (29) (41) (27)(29) and (28) to get
119875( sup119910le119909le119910+2120575
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 ge 4119872120598 + 119862radic119899ℎ4
119899)
le 119875( sup119910le119909le119910+2(119898
119899+1)119903119899
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 ge 4119872120598 + 119862radic119899ℎ4
119899)
le 119875(3 max1le119894le2(119898119899+1)
1003816100381610038161003816120572119899(119910 + 119894119903
119899) minus 120572
119899(119910)
1003816100381610038161003816
+119872radic119899119903119899+ 119862radic119899ℎ4
119899ge 4119872120598 + 119862radic119899ℎ4
119899)
ISRN Probability and Statistics 5
= 119875( max1le119894le2(119898
119899+1)
1003816100381610038161003816120572119899(119910 + 119894119903
119899) minus 120572
119899(119910)
1003816100381610038161003816 ge 119872120598)
le
119862119901min119901
11199012+1199013
(119872120598)119901
(2120575)min119901
11199012+1199013le 120578
(43)
for any fixed 119910 isin [0 1] Since the upper bound does notdepend on 119910 and the probability measure as well as supre-mum function are continuous we obtain
119875( sup0le119910le1minus2120575
sup119910le119909le119910+2120575
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 ge 4119872120598 + 119862radic119899ℎ4
119899) le 120578
(44)
where 4119872120598 + 119862radic119899ℎ4
119899is arbitrarily small
Since condition (10) is checked the proof is completed
3 Tightness of the Standard Empirical Process
In this section we deal with the standard empirical processbuilt on an associated sequence of uniformly [0 1]distributedrvrsquos 119880
119895119895ge1
that is
120572119899(119909) =
1
radic119899
119899
sum
119895=1
(119868 [119880119895le 119909] minus 119909) (45)
where 119909 isin [0 1] We shall relax the restrictions imposedon the process by Yu in [7] to obtain tightness Preciselywe do not need stationarity any more due to the techniquedrawn from [14] and we lower the assumed rate at which thecovariance tends to zero
While proving tightness of the empirical process we willuse the criterion proved in the first section as well as some ofour covariance inequalities
We shall start with the fact known under the name ofmultinomial theorem We recall it in the following lemma
Lemma 2 For natural numbers119898 119899 and 1199091 119909
2 119909
119898isin R
(1199091+ sdot sdot sdot + 119909
119899)119898
= sum
1198961+sdotsdotsdot+119896
119899=119898
(119898
1198961 119896
119899
) prod
1le119894le119899
119909119896119894
119894 (46)
where (119898
1198961 119896
119899) = 119898(119896
1 sdot sdot sdot 119896
119899) and 119896
1 119896
119899isin
0 1 119898
In particular
(1199091+ sdot sdot sdot + 119909
119899)4
le 4 sdot sum
1198961+sdotsdotsdot+119896
119899=4
1199091198961
1sdot sdot sdot 119909
119896119899
119899 (47)
which implies for 119878119899= sum
119899
119894=1119883
119894 that
1198641198784
119899le 4 sdot sum
1198961+sdotsdotsdot+119896
119899=4
119864 (1198831198961
1sdot sdot sdot 119883
119896119899
119899)
le 4 sdot
119899
sum
119905=1
sum
0le119894119895119896le119899minus1
10038161003816100381610038161003816119864 (119883
119905119883
119905+119894119883
119905+119894+119895119883
119905+119894+119895+119896)10038161003816100381610038161003816
(48)
Let us now introduce the following notation due to Doukhanand Louhichi (see [14]) for a sequence of centered rvrsquos119883
119905119899
119899ge1
119862119903119902
= sup100381610038161003816100381610038161003816Cov (119883
1199051
sdot sdot sdot 119883119905119898
119883119905119898+119903sdot sdot sdot 119883
119905119902
)100381610038161003816100381610038161003816 (49)
where the supremum runs over all divisions of the groupcomposed of 119902 rvrsquos into two subgroups such that the distancebetween the highest index of the rvrsquos in the first group and thelowest index of the rvrsquos from the second group is equal to 119903119903 isin 1 2 For 119903 = 0 we shall define 119862
0119902= 1 Let us then
put
1198831199051
= 119868 [1198801199051
le 119909] minus 119909 1198831199052
= 119868 [1198801199052
le 119909] minus 119909 (50)
We will now estimate the summands in (48)If max119894 119895 119896 = 119894 then
10038161003816100381610038161003816119864 119883
119905sdot (119883
119905+119894119883
119905+119894+119895119883
119905+119894+119895+119896)10038161003816100381610038161003816
=100381610038161003816100381610038161003816Cov (119883
119905 119883
119905+119894119883
119905+119894+119895119883
119905+119894+119895+119896)100381610038161003816100381610038161003816le 119862
1198944
(51)
Similarly for max119894 119895 119896 = 119896 we have
10038161003816100381610038161003816119864 (119883
119905119883
119905+119894119883
119905+119894+119895) sdot 119883
119905+119894+119895+11989610038161003816100381610038161003816
=10038161003816100381610038161003816Cov (119883
119905119883
119905+119894119883
119905+119894+119895 119883
119905+119894+119895+119896)10038161003816100381610038161003816le 119862
1198964
(52)
When max119894 119895 119896 = 119895 then
10038161003816100381610038161003816119864 (119883
119905119883
119905+119894) sdot (119883
119905+119894+119895119883
119905+119894+119895+119896)10038161003816100381610038161003816
=10038161003816100381610038161003816Cov (119883
119905119883
119905+119894 119883
119905+119894+119895119883
119905+119894+119895+119896)
+119864 (119883119905119883
119905+119894) 119864 (119883
119905+119894+119895119883
119905+119894+119895+119896)10038161003816100381610038161003816
le 1198621198954
+ 1198621198942119862
1198962
(53)
We keep on estimating the fourth moment of 119878119899by intro-
ducing 119905 isin 0 119899mdashthe index for which |119864(119883119905119883
119905+119894119883
119905+119894+119895
119883119905+119894+119895+119896
)| attains its maximum
1198641198784
119899le 4119899 sum
0le119894119895119896le119899minus1
10038161003816100381610038161003816119864 (119883
119905119883
119905+119894119883
119905+119894+119895119883
119905+119894+119895+119896)10038161003816100381610038161003816
=4119899( sum
0le119894119896le119895
[1198621198942sdot 119862
1198962+ 119862
1198954]+ sum
0le119894119895le119896
1198621198964+ sum
0le119896119895le119894
1198621198944)
= 4119899( sum
0le119894119896le119895
1198621198942sdot 119862
1198962+ 3 sum
0le119896119895le119894
1198621198944)
(54)
6 ISRN Probability and Statistics
The terms obtained in (54) may further be bounded fromabove in the following way
sum
0le119894119896le119895
1198621198942sdot 119862
1198962=
119899minus1
sum
119895=0
119895
sum
119894119896=0
1198621198942119862
1198962
le 119899
119899minus1
sum
119894119896=0
1198621198942119862
1198962= 119899(
119899minus1
sum
119894=0
1198621198942)
2
sum
0le119896119895le119894
1198621198944
=
119899minus1
sum
119894=0
119894
sum
119895119896=0
1198621198944
le
119899minus1
sum
119894=0
(119894 + 1) (119894 + 1) 1198621198944
=
119899minus1
sum
119894=0
(119894 + 1)2119862
1198944
(55)
We thus get the inequality
1198641198784
119899le 4([119899
119899minus1
sum
119894=0
1198621198942]
2
+ 3119899
119899minus1
sum
119894=0
(119894 + 1)2119862
1198944) (56)
We shall now focus on estimating 1198621199032
and 1198621199034 Since 119880
1199051
and 1198801199052
are associated uniformly [0 1] distributed rvrsquos 1198831199051
and 1198831199052
mdashas monotone functions of these rvrsquosmdashare associat-ed as well In order to bound
1198621199032
= sup0le1199051lt1199052le119899 1199052minus1199051=119903
Cov (1198831199051
1198831199052
) (57)
let us notice that from Schwarz inequality
Cov (1198831199051
1198831199052
) = 119864 (119868 [1198801199051
le 119909] 119868 [1198801199052
le 119909]) minus 1199092
le radic1199092 minus 1199092le 119909
(58)
On the other hand invoking inequalities from [13 15]
Cov (1198831199051
1198831199052
)
le sup119909isin[01]
[119875 (1198801199051
le 1199091198801199052
le 119909) minus 119875 (1198801199051
le 119909)119875 (1198801199052
le 119909)]
le 119862 sdot Cov13
(1198801199051
1198801199052
) for associated rvrsquos4 sdot Cov (119880
1199051
1198801199052
) for MTP2rvrsquos
(59)
As a consequence
1198621199032
le min 119909 119862 sdot Cov13
(1198801199051
1198801199052
) for associated rvrsquosmin 119909 4 sdot Cov (119880
1199051
1198801199052
) for MTP2rvrsquos
(60)
where 1199052minus 119905
1= 119903 Still we need the upper bound for 119862
1199034 It
turns out that119862
1199034
= sup1003816100381610038161003816100381610038161003816100381610038161003816
Cov2
prod
119894=1
(119868 [119880119905119894
le 119909] minus 119909)
4
prod
119894=3
(119868 [119880119905119894
le 119909] minus 119909)
1003816100381610038161003816100381610038161003816100381610038161003816
(61)
In other words among all divisions of the group 119883119905119894
119894 isin
1 2 3 4 into two subgroups 1198621199034
attains the biggest valuein case we take two subgroups consisted of two rvrsquos Thesupremum runs over the set 119905
1 119905
2 119905
3 119905
4isin N 0 le 119905
1lt 119905
2lt
1199053lt 119905
4le 119899 and 119905
3minus 119905
2= 119903 Elementary calculation leads to the
following formula
1198621199034
= sup 10038161003816100381610038161003816119875 (119880119905119894
le 119909 119894 isin 1 2 3 4)
minus 119875 (119880119905119894
le 119909 119894 isin 1 2) 119875 (119880119905119894
le 119909 119894 isin 3 4)
minus 119909 [119875 (119880119905119894
le 119909 119894 isin 1 2 3)
minus 119875 (119880119905119894
le 119909 119894 isin 1 2) 119875 (1198801199053
le 119909)
+ 119875 (119880119905119894
le 119909 119894 isin 1 3 4)
minus 119875 (119880119905119894
le 119909 119894 isin 3 4) 119875 (1198801199051
le 119909)
+ 119875 (119880119905119894
le 119909 119894 isin 1 2 4)
minus 119875 (119880119905119894
le 119909 119894 isin 1 2) 119875 (1198801199054
le 119909)
+ 119875 (119880119905119894
le 119909 119894 isin 2 3 4)
minus119875 (119880119905119894
le 119909 119894 isin 3 4) 119875 (1198801199052
le 119909)]
+ 1199092[119875 (119880
1199051
le 1199091198801199053
le 119909)
minus 119875 (1198801199051
le 119909)119875 (1198801199053
le 119909)
+ 119875 (1198801199051
le 1199091198801199054
le 119909)
minus 119875 (1198801199051
le 119909)119875 (1198801199054
le 119909)
+ 119875 (1198801199052
le 1199091198801199053
le 119909)
minus 119875 (1198801199052
le 119909)119875 (1198801199053
le 119909)
+ 119875 (1198801199052
le 1199091198801199054
le 119909)
minus119875 (1198801199052
le 119909)119875 (1198801199052
le 119909)]10038161003816100381610038161003816
le10038161003816100381610038161198771
1003816100381610038161003816 +10038161003816100381610038161198772
1003816100381610038161003816 +10038161003816100381610038161198773
1003816100381610038161003816
(62)
where for the sake of simplicity 1198771 119877
2 and 119877
3are respec-
tively the free coefficient the expression with 119909 and theexpression with 119909
2 Using the Lebowitz inequality (see [15]for instance) and inequalities obtained in [13 15] we arrive at10038161003816100381610038161198771
1003816100381610038161003816 =10038161003816100381610038161003816119875 (119880
119905119894
le 119909 119894 isin 1 2 3 4)
minus119875 (119880119905119894
le 119909 119894 isin 1 2) 119875 (119880119905119894
le 119909 119894 isin 3 4)10038161003816100381610038161003816
le 11986711988011990511198801199053
+ 11986711988011990511198801199054
+ 11986711988011990521198801199053
+ 11986711988011990521198801199054
ISRN Probability and Statistics 7
le
119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
)) for associated rvrsquos
4(
2
sum
119894=1
4
sum
119895=3
Cov (119880119905119894
119880119905119895
)) for MTP2rvrsquos
(63)
Analogously we get
10038161003816100381610038161198772
1003816100381610038161003816
le
2119909 sdot 119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
)) for associated rvrsquos
2119909 sdot 4(
2
sum
119894=1
4
sum
119895=3
Cov (119880119905119894
119880119905119895
)) for MTP2rvrsquos
10038161003816100381610038161198773
1003816100381610038161003816
le
1199092sdot 119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
) ) for associated rvrsquos
1199092sdot 4(
2
sum
119894=1
4
sum
119895=3
Cov (119880119905119894
119880119905119895
)) for MTP2rvrsquos
(64)
Eventually we have
1198621199034
le
119891 (119909) sdot119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
)) for associated rvrsquos
119891 (119909) sdot4(
2
sum
119894=1
4
sum
119895=3
Cov (119880119905119894
119880119905119895
)) for MTP2rvrsquos
(65)
where 1199053minus 119905
2= 119903 and 119891(119909) = (1 + 2119909 + 119909
2) for 119909 isin [0 1]
Let us now introduce the following notation
120579119903=
sup119905isinN
Cov (119880119905 119880
119905+119903) for 119903 isin 1 2
1 for 119903 = 0
(66)
and assume it decays powerly at rate 119886 in the following way
120579119903= 119874(
1
(119903 + 1)119886) for 119886 gt 0 119903 isin 0 1 (67)
Let us get back to inequality (56) we can now carry on Atfirst for associated rvrsquos
1198641198784
119899le 4[119899
119899minus1
sum
119903=0
min 119909 119862 sdot Cov13(119880
1199051
1198801199052
)]
2
+ 4 sdot 3119899
119899minus1
sum
119903=0
(119903 + 1)2119891 (119909) sdot 119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
))
le 119863 sdot ([119899
119899minus1
sum
119903=0
min 119909 (119903 + 1)minus1198863
]
2
+119899
119899minus1
sum
119903=0
(119903 + 1)2(119903 + 1)
minus1198863)
= 119863 sdot ([119899
119899
sum
119903=1
min 119909 119903minus1198863]
2
+ 119899
119899
sum
119903=1
1199032119903minus1198863
)
le 119863 sdot ([
[
119899 sum
119903lt119909minus3119886
119909 + 119899 sum
119903ge119909minus3119886
1
1199031198863]
]
2
+ 119899
119899
sum
119903=1
1199032minus1198863
)
le 1198631sdot (119899
2119909
2(119886minus3)119886+ 120585)
(68)
where119863 and1198631are constants and
120585 = 119899
119899
sum
119903=1
1199032minus1198863
=
119874(119899) 119886 gt 9
119874 (119899 ln 119899) 119886 = 9
119874 (1198994minus1198863
) 3 lt 119886 lt 9
(69)
It is worth mentioning that in the last inequality of (68) weused the estimate
int
infin
119909
1
119905119901119889119905 sim
1
119909119901minus1for 119901 gt 1 (70)
At the same time in the case of MTP2rvrsquos we get
1198641198784
119899le 119863
2sdot (119899
2119909
2(119886minus1)119886+ 120577) (71)
where1198632is constant and
120577 = 119899
119899
sum
119903=1
1199032minus119886
=
119874(119899) 119886 gt 3
119874 (119899 ln 119899) 119886 = 3
119874 (1198994minus119886
) 1 lt 119886 lt 3
(72)
Let now 120572119899(119909) = (1radic119899)sum
119899
119894=1(119868[119880
119905119894
le 119909] minus 119909) Then
120572119899(119909) minus 120572
119899(119910) =
1
radic119899
119899
sum
119894=1
(119868 [119909 lt 119880119905119894
le 119910] minus (119909 minus 119910))
for 119909 119910 isin [0 1]
(73)
8 ISRN Probability and Statistics
has the fourth moment estimatedmdashin the case of associatedrvrsquosmdashby
119864[120572119899(119909) minus 120572
119899(119910)]
4
= 119864[1
radic119899
119899
sum
119894=1
(119868 [119909 lt 119880119905119894
le 119910] minus (119909 minus 119910))]
4
=1
1198992119864[
119899
sum
119894=1
(119868 [119909 lt 119880119905119894
le 119910] minus (119909 minus 119910))]
4
le 1198631sdot (
1
119899211989921003816100381610038161003816119909 minus 119910
1003816100381610038161003816
2(119886minus3)119886
+120585
1198992)
= 1198631sdot (
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2(119886minus3)119886
+120585
1198992)
=
119874(119899minus1) 119886 gt 9
119874(ln 119899119899
) 119886 = 9
119874 (1198992minus1198863
) 3 lt 119886 lt 9
(74)
and in the case of MTP2rvrsquos by
119864[120572119899(119909) minus 120572
119899(119910)]
4
le 1198632sdot (
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2(119886minus1)119886
+120577
1198992)
=
119874(119899minus1) 119886 gt 3
119874(ln 119899119899
) 119886 = 3
119874 (1198992minus119886
) 1 lt 119886 lt 3
(75)
In light of the Shao and Yursquos criterion our process is tight forassociated rvrsquos when 119886 gt 6 and for MTP
2rvrsquos when 119886 gt 2
Let us sum up this result in the following theorem
Theorem 3 Let 120572119899(119909) = (1radic119899)sum
119899
119894=1(119868[119880
119894le 119909] minus 119909) be the
empirical process built on an associated sequence of uniformly[0 1] distributed rvrsquos 119880
119894119894ge1
Let also
120579119903= sup
119905isinNCov (119880
119905 119880
119905+119903) = 119874 ((119903 + 1)
minus119886)
where 119903 isin 0 1 119886 gt 0
(76)
Then 120572119899(119909)
119899ge1is tight for 119886 gt 6 If the rvrsquos 119880
119894119894ge1
are MTP2
then the process is tight for 119886 gt 2
Yu assumed stationarity of 119880119894119894ge1
andsum
infin
119899=111989965+]Cov(119880
0 119880
119899) lt infin for a positive constant ]
thus the rate of decay 119886 gt 7 5 Our result weakens consid-erably these assumptions especially in the case of MTP
2rvrsquos
Louhichi in [16] proposed a different tightness criterioninvolving the so-called bracketing numbers She managedto enhance Yursquos resultmdasheven more than Shao and Yu in[8]mdashsince she proved that it suffices to take 119886 gt 4 to gettightness of the empirical process based on the associated
rvrsquos Nevertheless she kept the assumption of stationarityvalid
In the final analysis our resultrsquos advantage is the absenceof the stationarity assumption and the rate of decay for120579119903remains (up to the authorrsquos knowledge) unimproved for
MTP2rvrsquos
Unfortunately with a view to obtainingweak convergenceof the process in question that is also convergence offinite-dimensional distributions we do not know how tomanage without the assumption of stationarityTherefore weconclude with the following corollary
Corollary 4 Let 120572119899(119909) = (1radic119899)sum
119899
119894=1(119868[119880
119894le 119909] minus 119909) be the
empirical process built on a stationary associated sequence ofuniformly [0 1] distributed rvrsquos 119880
119894119894ge1
Let also
120579119903= Cov (119880
1 119880
1+119903) = 119874 ((119903 + 1)
minus119886)
where 119903 isin 0 1 119886 gt 0
(77)
Then if 119886 gt 6
120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (78)
where 119861(sdot) is the zero mean Gaussian process on [0 1] withcovariance structure defined by
1205902(119909 119910) = 119909 and 119910 minus 119909119910
+
infin
sum
119895=1
[ Cov (119868 [1198801le 119909] 119868 [119880
119895+1le 119910])
+ Cov (119868 [1198801le 119910] 119868 [119880
119895+1le 119909])]
(79)
If the rvrsquos 119880119894119894ge1
are MTP2 then it suffices to be 119886 gt 2 in
order to claim the above convergence
Proof It remains to establish convergence of finite-dimen-sional distributions repeating the procedure from [7]
4 Tightness of the Kernel-TypeEmpirical Process
In this section we shall weaken assumption imposed on thecovariance structure of rvrsquos 119883
119895119895ge1
by Cai and Roussas in [1]for the kernel estimator of the df
119865119899(119909) =
1
119899
119899
sum
119895=1
119870(
119909 minus 119883119895
ℎ119899
) (80)
They deal with a stationary sequence of negatively associatedrvrsquos (cf [17]) and need the same condition as Yu [7] that is
1003816100381610038161003816Cov (1198831 119883
1+119899)1003816100381610038161003816 = 119874(
1
11989975+]) (81)
to get tightness of the smooth empirical process (see condi-tion (A4) in [1])
ISRN Probability and Statistics 9
It turns out that it suffices to have
1003816100381610038161003816Cov (1198801 119880
1+119899)1003816100381610038161003816 = 119874(
1
1198993119901(119901minus1)) (82)
where 119901 gt 4 is a positive constant taken from the tightnesscriterion (8) It is easy to see that asymptotically we get therate 3
On the way to prove it we will also take use of aRosenthal-type inequality due to Shao andYu (seeTheorem 2in [8]) we shall recall in the following lemma
Lemma 5 Let 119901 gt 2 and 119891 be a real valued function boundedby 1 with bounded first derivative Suppose that 119883
119899119899ge1
is asequence of stationary and associated rvrsquos such that for 119899 isin N
Cov (1198831 119883
119899) = 119874 (119899
minus119887) for some 119887 gt 119901 minus 1 (83)
Then for any 120583 gt 0 there exists some positive constant 119896120583
independent of the function 119891 for which
1198641003816100381610038161003816119878119899
(119891) minus 119864119878119899(119891)
1003816100381610038161003816
119901
le 119896120583(119899
1+12058310038171003817100381710038171003817119891
101584010038171003817100381710038171003817
2
+1198991199012 [
[
119899
sum
119895=1
10038161003816100381610038161003816Cov (119891 (119883
1)119891 (119883
119895))10038161003816100381610038161003816]
]
)
(84)
As we can see the lemma assumes association but itworks for negatively associated rvrsquos as well since in the proofit reaches back the result of Newman (see Proposition 15 in[18]) where both types of association are allowed
Let us recall that 120572119899(119909) = radic119899(119865
119899(119909) minus 119864119865
119899(119909)) where
119865119899(119909) = (1119899)sum
119899
119895=1119870((119909 minus 119883
119895)ℎ
119899) It is easy to see that
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
= 119899minus1199012
11986410038161003816100381610038161003816119878119899(119891 (119883
119895)) minus 119864119878
119899(119891 (119883
119895))10038161003816100381610038161003816
119901
(85)
where
119891 (119883119895) = 119870(
119909 minus 119883119895
ℎ119899
) minus 119870(
119910 minus 119883119895
ℎ119899
)
119878119899(119891) =
119899
sum
119895=1
119891 (119883119895)
(86)
With an intent to use Lemma 5 we need 119891 to be boundedfrom above by 1 (which in our case is obvious) and to havebounded 119891
1015840 thus we assume
sup119905isinR
10038161003816100381610038161003816119870
1015840(119905)
10038161003816100381610038161003816
1
ℎ119899
le119862
119870
ℎ119899
lt infin (87)
Now applying inequality (84) we have
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
le119896120583119899minus1199012
1198991+120583
1198622
119870
ℎ2
119899
+1198991199012[
[
119899
sum
119895=1
10038161003816100381610038161003816Cov (119891 (119883
1)119891 (119883
119895))10038161003816100381610038161003816]
]
1199012
(88)
Newman showed in [18] that
Cov (1198911(119883) 119891
2(119884))
= ∬R2119891
1015840
1(119909) 119891
1015840
2(119910)
times [119875 (119883 le 119909 119884 le 119910) minus 119875 (119883 le 119909) 119875 (119884 le 119910)] 119889119909 119889119910
(89)
if 1198911and 119891
2are real valued functions on R having square
integrable derivatives 1198911015840
1and 119891
1015840
2 respectively and provided
that 1198911(119883) 119891
2(119884) have finite secondmoments In light of that
equation and linearity of covariance we get10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
=
1003816100381610038161003816100381610038161003816∬
R2[119875 (119883
1lt 119909 minus ℎ
119899119904 119883
119895lt 119909 minus ℎ
119899119905)
minus119875 (1198831lt 119909 minus ℎ
119899119904 119883
119895lt 119910 minus ℎ
119899119905)]
minus [119875 (1198831lt 119909 minus ℎ
119899119904) 119875 (119883
119895lt 119909 minus ℎ
119899119905)
minus119875 (1198831lt 119909 minus ℎ
119899119904) 119875 (119883
119895lt 119910 minus ℎ
119899119905)]
minus [119875 (1198831lt 119910 minus ℎ
119899119904 119883
119895lt 119909 minus ℎ
119899119905)
minus119875 (1198831lt 119910 minus ℎ
119899119904 119883
119895lt 119910 minus ℎ
119899119905)]
minus [119875 (1198831lt 119910 minus ℎ
119899119904) 119875 (119883
119895lt 119910 minus ℎ
119899119905)
minus 119875 (1198831lt 119910 minus ℎ
119899119904)
times 119875 (119883119895lt 119909 minus ℎ
119899119905)] 119896 (119904) 119896 (119905) 119889119904 119889119905
10038161003816100381610038161003816100381610038161003816
(90)
Without the loss of generality we may and do assume that119909 gt 119910 thus10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
=
1003816100381610038161003816100381610038161003816∬
R2119875 (119910 minus ℎ
119899119904 le 119883
1lt 119909 minus ℎ
119899119904
119910 minus ℎ119899119905 le 119883
119895lt 119909 minus ℎ
119899119905)
minus 119875 (119910 minus ℎ119899119904 le 119883
1lt 119909 minus ℎ
119899119904)
times 119875 (119910 minus ℎ119899119905 le 119883
119895lt 119909 minus ℎ
119899119905) 119896 (119904) 119896 (119905) 119889119904 119889119905
1003816100381610038161003816100381610038161003816
(91)
and by triangle inequality we get10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
le ∬R2[int
119909minusℎ119899119904
119910minusℎ119899119904
int
119909minusℎ119899119905
119910minusℎ119899119905
1198911198831119883119895(119906 V) 119889119906 119889V
+int
119909minusℎ119899119904
119910minusℎ119899119904
1198911198831(119906) 119889119906int
119909minusℎ119899119905
119910minusℎ119899119905
119891119883119895(V) 119889V] 119896 (119904) 119896 (119905) 119889119904 119889119905
(92)
10 ISRN Probability and Statistics
1198911198831119883119895
(119906 V) 1198911198831
(119906) and 119891119883119895
(V) are the joint pdf of [1198831 119883
119895]
and marginal pdfrsquos of rvrsquos1198831and119883
119895 respectively We need
to further assume that 1198621stands for a common upper bound
of 1198911198831
and 1198911198831119883119895
for all 119895 isin N that is
100381710038171003817100381710038171198911198831
10038171003817100381710038171003817le 119862
1
1003817100381710038171003817100381710038171198911198831119883119895
100381710038171003817100381710038171003817le 119862
1forall119895 ge 1 (93)
Then we finally obtain
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816le 2119862
1
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
where = max 1198621 119862
2
1
(94)
On the other hand proceeding like Cai and Roussas in [1] wecan shortly get
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
le
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119909 minus 119883
1
ℎ119899
) 119870(
119909 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119909 minus 119883
1
ℎ119899
) 119870(
119910 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119910 minus 119883
1
ℎ119899
) 119870(
119909 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119910 minus 119883
1
ℎ119899
) 119870(
119910 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
le 41198622
10038161003816100381610038161003816Cov (119883
1 119883
119895)10038161003816100381610038161003816
13
(95)
where 1198622is a constant relevant to the covariance inequality
for negatively associated rvrsquos (see [13])We now arrive at the following inequality
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
le max 1198621 119862
2
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
10038161003816100381610038161003816Cov(119883
1 119883
119895)10038161003816100381610038161003816
13
(96)
Assuming |Cov(119891(1198831) 119891(119883
119895))| = 119874(119895
minus119903) and proceeding
similarly to (68) we can get
[
[
119899
sum
119895=1
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816]
]
1199012
le max 1198621 119862
2 [
[
sum
119895lt|119909minus119910|minus2(1199033)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
+ sum
119895ge|119909minus119910|minus2(1199033)
1
1198951199033]
]
1199012
le 2max 1198621 119862
21003816100381610038161003816119909 minus 119910
1003816100381610038161003816
((119903minus3)119903)119901
(97)
To sum up we obtain the following inequality
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
le 119896120583119899
1+120583minus1199012119862
2
119870
ℎ2
119899
+ 2max 1198621 119862
21003816100381610038161003816119909 minus 119910
1003816100381610038161003816
((119903minus3)119903)119901
le 1198621198991+120583minus1199012 1
ℎ2
119899
+1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
((119903minus3)119903)119901
(98)
where 119862 = 119896120583max1198622
119896 2max119862
1 119862
2 The formula of the
tightness criterion (8) implies that ((119903 minus 3)119903)119901 gt 1 so
119903 gt3119901
119901 minus 1 where 119901 gt 2 (99)
Let us recall the assumptions imposed on the bandwidhtsℎ
119899119899ge1
by Cai and Roussas in [1]B1 lim
119899rarrinfinℎ119899= 0 and ℎ
119899gt 0 for all 119899 isin N
+
B2 lim119899rarrinfin
119899ℎ119899= infin thus ℎ
119899= 119874(1119899
1minus120573) 120573 gt 0
B3 lim119899rarrinfin
119899ℎ4
119899= 0 hence ℎ
119899= 119874(1119899
14+120575) 120575 gt 0
In light of the above
1198991+120583minus1199012 1
ℎ2
119899
= 11989932+120583+2120575minus1199012
(100)
where 120583 gt 0 119901 gt 2 and 0 lt 120575 lt 34 Confrontation with thetightness criterion (8) forces
3
2+ 120583 + 2120575 minus
119901
2lt minus
1
2 (101)
which implies 119901 gt 4 Let us conclude with the followingtheorem
Theorem6 Let 120572119899(119909) = radic119899(119865
119899(119909)minus119864119865
119899(119909)) be the empirical
process built on the kernel estimator of the df119865 for a stationarysequence of negatively associated rvrsquos Assume that conditionsA1 A2 (87) B1 B2 B3 are satisfied Then provided that thetightness criterion (8) holds with 119901 gt 4 it suffices to demand
1003816100381610038161003816Cov (1198801 119880
1+119899)1003816100381610038161003816 = 119874(
1
1198993119901(119901minus1)) (102)
in order to obtain120572
119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (103)
where 119861(sdot) is the zero mean Gaussian process with covariancestructure defined by
1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)
+
infin
sum
119895=1
[ Cov (119868 [1198831le 119909] 119868 [119883
119895+1le 119910])
+ Cov (119868 [1198831le 119910] 119868 [119883
119895+1le 119909])]
(104)
and 119909 119910 isin [0 1]
Proof The remaining covergence of finite-dimensional dis-tributions of 120572
119899(119909) is established in [1]
ISRN Probability and Statistics 11
5 Weak Convergence of theRecursive Kernel-Type Empirical Processunder IID Assumption
The aim of this section is to show weak convergence of theempirical process
120572119899(119909) =
1
radic119899
119899
sum
119895=1
[119870(
119909 minus 119883119895
ℎ119895
) minus 119864119870(
119909 minus 119883119895
ℎ119895
)] (105)
built on iid rvrsquos 119883119895119895ge1
Wewill prove that120572
119899(sdot) convergesweakly to the Brownian
bridge 119861(sdot) with the following covariance structure
1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910) for 119909 119910 isin [0 1]
(106)
It turns out that the tightness criterion (8) suffices to reachthe goal Convergence of finite-dimensional distributions of120572
119899(sdot) holds and we shall show itProceeding like Cai and Roussas in [1] we need to show
that for any 119886 119887 isin R
119886120572119899(119909) + 119887120572
119899(119910) 997888rarr 119886119861 (119909) + 119887119861 (119910) in distribution
(107)
Let us introduce the notation
119884119894(119909) = 119870(
119909 minus 119883119894
ℎ119894
) minus 119864119870(119909 minus 119883
119894
ℎ119894
) (108)
and look into the covariance structure of 120572119899(sdot)
Cov (120572119899(119909) 120572
119899(119910))
= Cov( 1
radic119899
119899
sum
119894=1
119884119894(119909)
1
radic119899
119899
sum
119894=1
119884119894(119910))
=1
119899
119899
sum
119894=1
Cov (119884119894(119909) 119884
119894(119910))
=1
119899
119899
sum
119894=1
Cov(119870(119909 minus 119883
119894
ℎ119894
) 119870(119910 minus 119883
119894
ℎ119894
))
(109)
where the second equality follows from assumed indepen-dence of rvrsquos 119883
119894119894ge1
Firstly we observe that each summand converges to119865(119909and
119910) minus 119865(119909)119865(119910) since
Cov(119870(119909 minus 119883
119894
ℎ119894
) 119870(119910 minus 119883
119894
ℎ119894
))
= 119864[119870(119909 minus 119883
119894
ℎ119894
)119870(119910 minus 119883
119894
ℎ119894
)]
minus 119864119870(119909 minus 119883
119894
ℎ119894
)119864119870(119910 minus 119883
119894
ℎ119894
)
(110)
where
119864[119870(119909 minus 119883
119894
ℎ119894
)119870(119910 minus 119883
119894
ℎ119894
)]
= int
119909and119910
minusinfin
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
+ int
119909or119910
119909and119910
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
+ int
infin
119909or119910
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
(111)
and 119865 denotes the common df of the rvrsquos 119883119894119894ge1
In [6] itwas shown that
119864119870(119909 minus 119883
119894
ℎ119894
) 997888rarr 119865 (119909) (112)
and recalling that the kernel function 119870 is a df as well wearrive at the conclusion
Secondly applying Toeplitz lemma we get
Cov (120572119899(119909) 120572
119899(119910)) 997888rarr 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)
= 1205902(119909 119910) 119899 997888rarr infin
(113)
Thus
Var (119886120572119899(119909) + 119887120572
119899(119910))
= 1198862 Var (120572
119899(119909)) + 2119886119887Cov (120572
119899(119909) 120572
119899(119910))
+ 1198872 Var (120572
119899(119910))
997888rarr 1198862120590
2(119909 119909) + 2119886119887120590
2(119909 119910) + 119887
2120590
2(119910 119910) 119899 997888rarr infin
(114)
We are now on the way to prove that 119886120572119899(119909) + 119887120572
119899(119910) con-
verges in distribution to 119886119861(119909) + 119887119861(119910) sim N(0 1198862120590
2(119909 119909) +
21198861198871205902(119909 119910) + 119887
2120590
2(119910 119910)) which in other words means that
1
radic119899
infin
sum
119894=1
119886119884119894(119909) + 119887119884
119894(119910)
radicVar (119886120572119899(119909) + 119887120572
119899(119910))
997888rarr N (0 1) (115)
In order to obtain (115) we will use Lyapunov condition
lim119899rarrinfin
1
1198632+120575
119899
119899
sum
119894=1
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884119894(119910)
1003816100381610038161003816
2+120575
= 0 (116)
for 120575 = 1 where 1198632
119899= Var(sum119899
119894=1[119886119884
119894(119909) + 119887119884
119894(119910)]) Since
|119870(sdot) minus 119864119870(sdot)| le 2
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884i (119910)1003816100381610038161003816
3
le 8 (1198863+ 119887
3) (117)
Moreover 1198632
119899= 119899Var(119886120572
119899(119909) + 119887120572
119899(119910)) together with (114)
yields
1198633
119899= 119874 (119899
32) (118)
12 ISRN Probability and Statistics
Combining (117) with (118) we obtain
1
1198633
119899
119899
sum
119894=1
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884119894(119910)
1003816100381610038161003816
3
= 119874 (119899minus12
) (119)
which shows that Lyapunov condition holds and completesthe proof of convergence of finite-dimensional distributionsof 120572
119899(119909)We summarize the result of that section in the following
theorem
Theorem 7 Let 120572119899(119909) be the recursive kernel-type empirical
process defined by (105) built on iid rvrsquos 119883119895119895ge1
withmarginal df 119865 If 120572
119899(sdot) satisfies the tightness criterion (8) then
120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (120)
where 119861 is the Brownian bridge
Acknowledgment
The author would like to thank the referees for the carefulreading of the paper and helpful remarks
References
[1] Z Cai and G G Roussas ldquoWeak convergence for smoothestimator of a distribution function under negative associationrdquoStochastic Analysis and Applications vol 17 no 2 pp 145ndash1681999
[2] Z Cai andG G Roussas ldquoEfficient Estimation of a DistributionFunction under Qudrant Dependencerdquo Scandinavian Journal ofStatistics vol 25 no 1 pp 211ndash224 1998
[3] ZW Cai and G G Roussas ldquoUniform strong estimation under120572-mixing with ratesrdquo Statistics amp Probability Letters vol 15 no1 pp 47ndash55 1992
[4] G G Roussas ldquoKernel estimates under association stronguniform consistencyrdquo Statistics amp Probability Letters vol 12 no5 pp 393ndash403 1991
[5] Y F Li and S C Yang ldquoUniformly asymptotic normalityof the smooth estimation of the distribution function underassociated samplesrdquo Acta Mathematicae Applicatae Sinica vol28 no 4 pp 639ndash651 2005
[6] Y Li C Wei and S Yang ldquoThe recursive kernel distributionfunction estimator based onnegatively and positively associatedsequencesrdquo Communications in Statistics vol 39 no 20 pp3585ndash3595 2010
[7] H Yu ldquoA Glivenko-Cantelli lemma and weak convergence forempirical processes of associated sequencesrdquo ProbabilityTheoryand Related Fields vol 95 no 3 pp 357ndash370 1993
[8] Q M Shao and H Yu ldquoWeak convergence for weightedempirical processes of dependent sequencesrdquo The Annals ofProbability vol 24 no 4 pp 2098ndash2127 1996
[9] P Billingsley Convergence of Probability Measures Wiley Seriesin Probability and Statistics Probability and Statistics JohnWiley amp Sons New York NY USA 1999
[10] J D Esary F Proschan and D W Walkup ldquoAssociation ofrandom variables with applicationsrdquo Annals of MathematicalStatistics vol 38 pp 1466ndash1474 1967
[11] A Bulinski and A Shashkin Limit Theorems for AssociatedRandom Fields and Related Systems vol 10 of Advanced Serieson Statistical Science and Applied Probability World Scientific2007
[12] S Karlin and Y Rinott ldquoClasses of orderings of measures andrelated correlation inequalities I Multivariate totally positivedistributionsrdquo Journal of Multivariate Analysis vol 10 no 4 pp467ndash498 1980
[13] P Matuła and M Ziemba ldquoGeneralized covariance inequali-tiesrdquo Central European Journal of Mathematics vol 9 no 2 pp281ndash293 2011
[14] P Doukhan and S Louhichi ldquoA new weak dependence condi-tion and applications to moment inequalitiesrdquo Stochastic Proc-esses and their Applications vol 84 no 2 pp 313ndash342 1999
[15] P Matuła ldquoA note on some inequalities for certain classes ofpositively dependent random variablesrdquo Probability and Math-ematical Statistics vol 24 no 1 pp 17ndash26 2004
[16] S Louhichi ldquoWeak convergence for empirical processes of asso-ciated sequencesrdquo Annales de lrsquoInstitut Henri Poincare vol 36no 5 pp 547ndash567 2000
[17] K Joag-Dev and F Proschan ldquoNegative association of randomvariables with applicationsrdquoThe Annals of Statistics vol 11 no1 pp 286ndash295 1983
[18] C M Newman ldquoAsymptotic independence and limit theoremsfor positively and negatively dependent random variablesrdquo inInequalities in Statistics and Probability Y L Tong Ed vol 5pp 127ndash140 Institute ofMathematical StatisticsHayward CalifUSA 1984
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 ISRN Probability and Statistics
In this paper we shall study the empirical process 120572119899(119909)
generated by the generalized kernel estimator of the df givenby the formula
119865119899(119909) =
1
119899
119899
sum
119895=1
119870(
119909 minus 119883119895
ℎ119899119895
) where (5)
A1 119883119895119895ge1
is a sequence of absolutely continuous id rvrsquostaking values in [0 1] and having twice differentiabledf 119865 with first and second derivative bounded
A2 119870 is a kernel function such that intR 119906119889119870(119906) = 0 andintR 119906
2119889119870(119906) lt infin with bounded derivative 119896
A3 ℎ119899119895119899ge1119895isin1119899
is a sequence of positive constantssubject to the following conditions lim
119895119899rarrinfinℎ119899119895
= 0lim
119895119899rarrinfin119899ℎ
4
119899119895= 0 (actually since 119895 le 119899 119895 rarr infin
under the limit suffices)
Explicitly we take a look onto the process
120572119899(119909) =
1
radic119899sdot
119899
sum
119895=1
[119870(
119909 minus 119883119895
ℎ119899119895
) minus 119864119870(
119909 minus 119883119895
ℎ119899119895
)] (6)
we shall call from now on the generalized empirical processLet us pay attention to the fact that in the case of
(i) ℎ119899119895
= ℎ119899for 119895 isin 1 119899 120572
119899(119909) is the empirical
process based on the kernel estimator of the df 119865(ii) ℎ
119895119895= ℎ
119895+1119895= ℎ
119895+2119895= sdot sdot sdot = ℎ
119895for 119895 isin N 120572
119899(119909)
is the empirical process based on the recursive kernelestimator of the df 119865
(iii) 119870(119905) = 119868[0infin]
(119905) 120572119899(119909) is the standard empirical
process (based on the empirical df)
It is well known that the crucial procedure in showingweak convergence for an empirical process is to verify tight-ness In [8] Shao and Yu gave the following criterion
exist1198621gt0exist
119901gt2exist
1199011gt1exist
0le1199031le1exist
1199012gt1minus1199031
forall119909119910isin[01]
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
le 1198621(1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
1199011
+ 119899minus119901221003816100381610038161003816119909 minus 119910
1003816100381610038161003816
1199031
)
(7)
under which the standard empirical process based on sta-tionary sequence of uniform [0 1] rvrsquos is tight It is statedthere that the proof of that fact is an easy standard procedureparallel to the one presented in [9] It is the main aim of thispaper to carry it in details but for the generalized empiricalprocess defined by (6) and without assuming stationarityNevertheless wewill always return to stationarity assumptionwhile establishing weak convergence
In order to obtain tightness one has to assume appro-priate covariance structure of the underlying rvrsquos that isthe covariance of a pair of rvrsquos 119883
119894and 119883
119895has to decline
at the right rate while 119894 and 119895 are growing apart In thispaper we lower the demanded rate of covariance decay usingthe covariance inequalities for associated (cf [10 11]) andmultivariate totally positive of order 2 (MTP
2) (cf [12]) rvrsquos
obtained in [13]
The paper is organized as follows In Section 2 we presentthe proof of the Shao and Yursquos tightness criterion formulatedfor our generalized empirical process Sections 3 and 4 aredevoted to application of the criterion to showing tightnessand thus weak convergence of the specific types of empiricalprocesses Section 5 concerns weak convergence of the recur-sive kernel-type process for iid rvrsquos
2 Tightness Criterion
We start with the key point of the paper
Theorem 1 Let 120572119899(119909)
119899ge1isin 119863[0 1] be the generalized empir-
ical process defined as in (6) One assume that A1 A2 A3 holdIf there exist constants 119862 gt 0 119901 gt 2 119901
1gt 1 0 le 119901
3le 1
1199012gt 1 minus 119901
3 such that for any 119909 119910 isin [0 1] and 119899 isin N the
following inequality holds
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
le 119862 sdot (1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
1199011
+ 119899minus11990122sdot1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
1199013
)
(8)
then the process 120572119899(119909)
119899ge1is tight in119863[0 1] with 119869
1topology
Proof The proof boils down to showing that under theassumptions made inTheorem 1 conditions ofTheorem 132in [9] hold Let us recall that in light of the above mentionedtheorem a process 120572
119899(119909)
119899ge1is tight in119863[0 1] if
lim119886rarrinfin
lim sup119899rarrinfin
119875 (1003817100381710038171003817120572119899
1003817100381710038171003817 ge 119886) = 0 (9)
lim120575rarr0
lim sup119899rarrinfin
119875 (1199081015840
1(120572
119899 120575) ge 120598) = 0 forall120598 gt 0 (10)
where sdot is the supremum norm that is1003817100381710038171003817120572119899
1003817100381710038171003817 = sup0le119909le1
1003816100381610038161003816120572119899(119909)
1003816100381610038161003816 (11)
and 1199081015840
1(120572
119899 120575) is the modulus of continuity of the function
120572119899isin 119863[0 1] that is
1199081015840
1(120572
119899 120575) = inf
119905119894
max1le119894le]
sup119909119910isin[119905119894minus1119905119894)
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 (12)
The infimum runs over all finite ldquo120575-sparserdquo decompositions[119905
119894minus1 119905
119894) 119894 isin 1 ] of the interval [0 1] In other words it
runs over all choices of increasingly ordered points 1199051198941le119894le]
such that min1le119894le](119905119894 minus 119905
119894minus1) gt 120575 where 119905
0= 0 119905] = 1
Let us first show that condition (9) is satisfied Accordingto the corollary following Theorem 132 in [9] it suffices toshow
lim119886rarrinfin
lim sup119899rarrinfin
119875 (1205721198991003816100381610038161003816120572119899
(119909)1003816100381610038161003816 ge 119886) = 0 forall119909 isin [0 1] (13)
that is
forall120578ge0
exist119886gt0
forall119899isinN119875 (120572
1198991003816100381610038161003816120572119899
(119909)1003816100381610038161003816 ge 119886) le 120578 forall119909 isin [0 1] (14)
Let us fix 119909 isin [0 1] and 120578 gt 0 We need to find 119886 = 119886(120578) gt 0such that
119875 (radic11989910038161003816100381610038161003816119865119899(119909) minus 119864119865
119899(119909)
10038161003816100381610038161003816ge 119886) le 120578 forall119899 isin N (15)
ISRN Probability and Statistics 3
where
119865119899(119909) =
1
119899
119899
sum
119895=1
119870(
119909 minus 119883119895
ℎ119899119895
) (16)
is the generalized kernel estimator of the df119865 ApplyingChe-byshevrsquos inequality we shall find 119886 satisfying
119886 geradic119899
120578sdot 119864
10038161003816100381610038161003816119865119899(119909) minus 119864119865
119899(119909)
10038161003816100381610038161003816forall119899 isin N (17)
Such 119886 exists if 119864|120572119899(119909)| lt infin for all 119899 ge 1 Implementing
119910 = 0 and 1199013= 0 in the assumption (8) we get for 119901 gt 2
1199011 119901
2gt 1
1198641003816100381610038161003816120572119899
(119909)1003816100381610038161003816
119901
le 119862 sdot(|119909|1199011+119899
minus11990122|119909|
1199013) le 119862 sdot(1+
1
11989911990122) le 2119862
(18)
Now applying Holderrsquos inequality we arrive at
1198641003816100381610038161003816120572119899
(119909)1003816100381610038161003816 le
4radic119864
1003816100381610038161003816120572119899(119909)
1003816100381610038161003816
4
(19)
which in light of the inequality (18) for 119901 = 4 is boundedfrom above by 4radic2119862 lt infin Therefore condition (9) holds
We shall now proceed to checking condition (10) Let usrecall that the modulus of continuity of the function 119909 isin
119862[0 1] 120575 gt 0 is given by the formula
1199081(119909 120575) = sup
0le119910le1minus120575
sup119910le119909le119910+120575
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 (20)
As it is shown in [9] (page 123) there is a relation between119908
1(sdot sdot) and 1199081015840
1(sdot sdot) in the spaces 119862[0 1] and119863[0 1] relatively
Namely for 119909 isin 119863[0 1] and 120575 lt 12
1199081(119909 2120575) ge 119908
1015840
1(119909 120575) (21)
Thus
119875 (1199081015840
1(120572
119899 120575) ge 120598) le 119875 (119908
1(120572
119899 2120575) ge 120598) (22)
Therefore condition (10) holds if we show
forall120598gt0
forall120578gt0
exist0le120575lt12
exist1198990
forall119899ge1198990
119875 (1199081(120572
119899 2120575) ge 120598) le 120578 (23)
With a view to obtaining the desired inequality we shall pro-ceed patiently in five steps
Step 1 We need a moment inequality for the rv 120572119899(119909)minus120572
119899(119910)
involving the distance between the points 119909 and 119910 Let usthen assume that for the constants 119862 119901 119901
1 119901
2 119901
3given in
Theorem 1 inequality (8) holds Fix 120598 gt 0 120578 gt 0 and definethe quantity 119903
119899= 120598radic119899 119899 isin N Next we fix 119909 119910 isin [0 1] and
take 119899 large enough so that 119903119899le |119909minus119910|Then 119899minus12
le |119909minus119910|120598minus1
and we have
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
le 119862 (1 + 120598minus1199012)1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
min11990111199012+1199013
(24)
Step 2 Let us now fix 119899 isin N 120575 isin (0 1) and consider the fol-lowing rvrsquos
120594119894= 120572
119899(119910 + 119894119903
119899) minus 120572
119899(119910 + (119894 minus 1) 119903
119899) (25)
for 119894 isin 1 2 119898119899 where119898
119899= 119898(119899 120575) is such that 119903
119899119898
119899le 120575
It is easy to see that for 119878119894= sum
119894
119896=1120594
119896
max1le119894le119898
119899
1003816100381610038161003816119878119894
1003816100381610038161003816 = max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119903
119899) minus 120572
119899(119910)
1003816100381610038161003816 (26)
Let us notice that for rvrsquos 120594119894119894ge1
the conditions of Theo-rem 102 in [9] are satisfied with 4120573 = 119901 119906
119897= 119903
119899forall
119894lt119897le119895and
2120572 = min1199011 119901
2+ 119901
3 Therefore we are equipped with the
following maximal inequality
119875( max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119903
119899) minus 120572
119899(119910)
1003816100381610038161003816 ge 120582)
le
119862119901min119901
11199012+1199013
120582119901(119898
119899119903119899)min119901
11199012+1199013
(27)
for all 120582 gt 0
Step 3 Let 119872 = sup119909isin[01]
119891(119909) where 119891 is the pdf of rvrsquos119883
119895119895ge1
For fixed 120598 120578 gt 0 let us take 120575 gt 0 such that
119862119901min119901
11199012+1199013
(119872120598)119901
sdot (2120575)min119901
11199012+1199013lt 120578 (28)
and define119898119899= lfloor120575119903
119899rfloor For sufficiently large 119899 isin N we have
119898119899ge 1 and then we get
119903119899119898
119899le 120575 lt (119898
119899+ 1) 119903
119899le 2119898
119899119903119899le 120575 (29)
Step 4 Our goal is to obtain an inequality which enables usto bound the supremum of the increment of the function 120572
119899
via the maximum of the increments of that function on somesubintervals To bemore precise wewill find the upper boundfor
sup119910le119909le119910+119898
119899119902
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 (30)
in terms of
max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572
119899(119910)
1003816100381610038161003816 (31)
where 119909 119910 isin [0 1] 119910 le 119909 le 119910 + 119902 119902 ge 120598119899 and119898119899is defined
as in Step 3Let us recall that 120572
119899(119909) = radic119899(119865
119899(119909) minus 119864119865
119899(119909)) where
119865119899(119909) = (1119899)sum
119899
119895=1119870((119909 minus 119883
119895)ℎ
119899119895) From the triangle ine-
quality we can see that
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816
leradic119899
119899
10038161003816100381610038161003816100381610038161003816100381610038161003816
119899
sum
119895=1
[119870(
119909 minus 119883119895
ℎ119899119895
) minus 119870(
119910 minus 119883119895
ℎ119899119895
)]
10038161003816100381610038161003816100381610038161003816100381610038161003816
+ radic119899 (10038161003816100381610038161003816119864119865
119899(119909) minus 119865 (119909)
10038161003816100381610038161003816+10038161003816100381610038161003816119864119865
119899(119910) minus 119865 (119910)
10038161003816100381610038161003816
+1003816100381610038161003816119865 (119909) minus 119865 (119910)
1003816100381610038161003816 )
(32)
4 ISRN Probability and Statistics
Since 119870 is a nondecreasing function and applying triangleinequality again we get
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816
le1003816100381610038161003816120572119899
(119910 + 119902) minus 120572119899(119910)
1003816100381610038161003816 +radic119899
10038161003816100381610038161003816119864119865
119899(119910 + 119902) minus 119864119865
119899(119910)
10038161003816100381610038161003816
+ radic119899 (10038161003816100381610038161003816119864119865
119899(119909) minus 119865 (119909)
10038161003816100381610038161003816+10038161003816100381610038161003816119864119865
119899(119910) minus 119865 (119910)
10038161003816100381610038161003816
+1003816100381610038161003816119865 (119909) minus 119865 (119910)
1003816100381610038161003816 )
(33)
Cai and Roussas in [1] showed that under assumptions madeon the df 119865 and the kernel function119870 we have
10038161003816100381610038161003816119864119865
119899(119909) minus 119865 (119909)
10038161003816100381610038161003816= 119874 (ℎ
2
119899) (34)
Similarly by Taylor expansion it is easy to see that
1003816100381610038161003816119865 (119909) minus 119865 (119910)1003816100381610038161003816 le
10038171003817100381710038171198911003817100381710038171003817 sdot
1003816100381610038161003816119909 minus 1199101003816100381610038161003816 (35)
Thus
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 le1003816100381610038161003816120572119899
(119910 + 119902) minus 120572119899(119910)
1003816100381610038161003816
+ 119862radic119899ℎ4
119899+119872radic119899119902
(36)
where 119872 = sup119909isin[01]
119891(119909) and 119862 = 1198621198911015840119870
is a positive con-stant dependent on the functions 119865 and119870
Let us recall that 119910 le 119909 le 119910+ 119902119898119899= lfloor120575119903
119899rfloor and 119902 ge 120598119899
Taking 119902 = (120598radic119899)(= 119903119899) we notice that 119898
119899119902 = lfloor120575119903
119899rfloor119903
119899le 120575
thus when 119899 rarr infin the interval [119910 119910 + 119898119899119902] coincides with
[119910 119910 + 120575] Approaching the aim of Step 4 let us observe that
sup119910le119909le119910+119898
119899119902
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 = max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572
119899(119910)
1003816100381610038161003816
+1003816100381610038161003816120572119899
(1199090) minus 120572
119899(119910 + 119894max119902)
1003816100381610038161003816
(37)
where 1199090isin [119910+119894
0119902 119910+(119894
0+1)119902] is a point at which the above
supremum is attained and
119894max =
1198940+ 1 if 1003816100381610038161003816120572119899
(119910 + [1198940+ 1] 119902) minus 120572
119899(119910)
1003816100381610038161003816
ge1003816100381610038161003816120572119899
(119910 + 1198940119902) minus 120572
119899(119910)
1003816100381610038161003816
1198940
otherwise(38)
Without the loss of generality we may assume that 119894max = 1198940+
1 which implies
sup119910le119909le119910+119898
119899119902
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 = max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572
119899(119910)
1003816100381610038161003816
+1003816100381610038161003816120572119899
(1199090) minus 120572
119899(119910 + 119894
0119902)1003816100381610038161003816
(39)
Now applying inequality (36) the definition of 119894max and thetriangle inequality we have
sup119910le119909le119910+119898
119899119902
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816
le max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572
119899(119910)
1003816100381610038161003816
+1003816100381610038161003816120572119899
(119910 + [1198940+ 1] 119902) minus 120572
119899(119910 + 119894
0119902)1003816100381610038161003816
+ 119872radic119899119902 + 119862radic119899ℎ4
119899
le max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572
119899(119910)
1003816100381610038161003816
+1003816100381610038161003816120572119899
(119910 + [1198940+ 1] 119902) minus 120572
119899(119910)
1003816100381610038161003816
+1003816100381610038161003816120572119899
(119910 + 1198940119902) minus 120572
119899(119910)
1003816100381610038161003816 + 119872radic119899119902 + 119862radic119899ℎ4
119899
le 3 max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572
119899(119910)
1003816100381610038161003816 + 119872radic119899119902 + 119862radic119899ℎ4
119899
(40)
If we plug in 119902 = 119903119899we arrive at
sup119910le119909le119910+119898
119899119903119899
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 le 3 max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119903
119899) minus 120572
119899(119910)
1003816100381610038161003816
+ 119872radic119899119903119899+ 119862radic119899ℎ4
119899
(41)
Step 5 Finally we are in a position to obtain inequality (23)that is
forall120598gt0forall
120578gt0exist
0le120575lt12exist
1198990
forall119899ge1198990
119875( sup0le119910le1minus2120575
sup119910le119909le119910+2120575
1003816100381610038161003816120572119899(119909)minus120572
119899(119910)
1003816100381610038161003816
ge 120598) le 120578
(42)
We now successivelymake use the inequalities (29) (41) (27)(29) and (28) to get
119875( sup119910le119909le119910+2120575
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 ge 4119872120598 + 119862radic119899ℎ4
119899)
le 119875( sup119910le119909le119910+2(119898
119899+1)119903119899
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 ge 4119872120598 + 119862radic119899ℎ4
119899)
le 119875(3 max1le119894le2(119898119899+1)
1003816100381610038161003816120572119899(119910 + 119894119903
119899) minus 120572
119899(119910)
1003816100381610038161003816
+119872radic119899119903119899+ 119862radic119899ℎ4
119899ge 4119872120598 + 119862radic119899ℎ4
119899)
ISRN Probability and Statistics 5
= 119875( max1le119894le2(119898
119899+1)
1003816100381610038161003816120572119899(119910 + 119894119903
119899) minus 120572
119899(119910)
1003816100381610038161003816 ge 119872120598)
le
119862119901min119901
11199012+1199013
(119872120598)119901
(2120575)min119901
11199012+1199013le 120578
(43)
for any fixed 119910 isin [0 1] Since the upper bound does notdepend on 119910 and the probability measure as well as supre-mum function are continuous we obtain
119875( sup0le119910le1minus2120575
sup119910le119909le119910+2120575
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 ge 4119872120598 + 119862radic119899ℎ4
119899) le 120578
(44)
where 4119872120598 + 119862radic119899ℎ4
119899is arbitrarily small
Since condition (10) is checked the proof is completed
3 Tightness of the Standard Empirical Process
In this section we deal with the standard empirical processbuilt on an associated sequence of uniformly [0 1]distributedrvrsquos 119880
119895119895ge1
that is
120572119899(119909) =
1
radic119899
119899
sum
119895=1
(119868 [119880119895le 119909] minus 119909) (45)
where 119909 isin [0 1] We shall relax the restrictions imposedon the process by Yu in [7] to obtain tightness Preciselywe do not need stationarity any more due to the techniquedrawn from [14] and we lower the assumed rate at which thecovariance tends to zero
While proving tightness of the empirical process we willuse the criterion proved in the first section as well as some ofour covariance inequalities
We shall start with the fact known under the name ofmultinomial theorem We recall it in the following lemma
Lemma 2 For natural numbers119898 119899 and 1199091 119909
2 119909
119898isin R
(1199091+ sdot sdot sdot + 119909
119899)119898
= sum
1198961+sdotsdotsdot+119896
119899=119898
(119898
1198961 119896
119899
) prod
1le119894le119899
119909119896119894
119894 (46)
where (119898
1198961 119896
119899) = 119898(119896
1 sdot sdot sdot 119896
119899) and 119896
1 119896
119899isin
0 1 119898
In particular
(1199091+ sdot sdot sdot + 119909
119899)4
le 4 sdot sum
1198961+sdotsdotsdot+119896
119899=4
1199091198961
1sdot sdot sdot 119909
119896119899
119899 (47)
which implies for 119878119899= sum
119899
119894=1119883
119894 that
1198641198784
119899le 4 sdot sum
1198961+sdotsdotsdot+119896
119899=4
119864 (1198831198961
1sdot sdot sdot 119883
119896119899
119899)
le 4 sdot
119899
sum
119905=1
sum
0le119894119895119896le119899minus1
10038161003816100381610038161003816119864 (119883
119905119883
119905+119894119883
119905+119894+119895119883
119905+119894+119895+119896)10038161003816100381610038161003816
(48)
Let us now introduce the following notation due to Doukhanand Louhichi (see [14]) for a sequence of centered rvrsquos119883
119905119899
119899ge1
119862119903119902
= sup100381610038161003816100381610038161003816Cov (119883
1199051
sdot sdot sdot 119883119905119898
119883119905119898+119903sdot sdot sdot 119883
119905119902
)100381610038161003816100381610038161003816 (49)
where the supremum runs over all divisions of the groupcomposed of 119902 rvrsquos into two subgroups such that the distancebetween the highest index of the rvrsquos in the first group and thelowest index of the rvrsquos from the second group is equal to 119903119903 isin 1 2 For 119903 = 0 we shall define 119862
0119902= 1 Let us then
put
1198831199051
= 119868 [1198801199051
le 119909] minus 119909 1198831199052
= 119868 [1198801199052
le 119909] minus 119909 (50)
We will now estimate the summands in (48)If max119894 119895 119896 = 119894 then
10038161003816100381610038161003816119864 119883
119905sdot (119883
119905+119894119883
119905+119894+119895119883
119905+119894+119895+119896)10038161003816100381610038161003816
=100381610038161003816100381610038161003816Cov (119883
119905 119883
119905+119894119883
119905+119894+119895119883
119905+119894+119895+119896)100381610038161003816100381610038161003816le 119862
1198944
(51)
Similarly for max119894 119895 119896 = 119896 we have
10038161003816100381610038161003816119864 (119883
119905119883
119905+119894119883
119905+119894+119895) sdot 119883
119905+119894+119895+11989610038161003816100381610038161003816
=10038161003816100381610038161003816Cov (119883
119905119883
119905+119894119883
119905+119894+119895 119883
119905+119894+119895+119896)10038161003816100381610038161003816le 119862
1198964
(52)
When max119894 119895 119896 = 119895 then
10038161003816100381610038161003816119864 (119883
119905119883
119905+119894) sdot (119883
119905+119894+119895119883
119905+119894+119895+119896)10038161003816100381610038161003816
=10038161003816100381610038161003816Cov (119883
119905119883
119905+119894 119883
119905+119894+119895119883
119905+119894+119895+119896)
+119864 (119883119905119883
119905+119894) 119864 (119883
119905+119894+119895119883
119905+119894+119895+119896)10038161003816100381610038161003816
le 1198621198954
+ 1198621198942119862
1198962
(53)
We keep on estimating the fourth moment of 119878119899by intro-
ducing 119905 isin 0 119899mdashthe index for which |119864(119883119905119883
119905+119894119883
119905+119894+119895
119883119905+119894+119895+119896
)| attains its maximum
1198641198784
119899le 4119899 sum
0le119894119895119896le119899minus1
10038161003816100381610038161003816119864 (119883
119905119883
119905+119894119883
119905+119894+119895119883
119905+119894+119895+119896)10038161003816100381610038161003816
=4119899( sum
0le119894119896le119895
[1198621198942sdot 119862
1198962+ 119862
1198954]+ sum
0le119894119895le119896
1198621198964+ sum
0le119896119895le119894
1198621198944)
= 4119899( sum
0le119894119896le119895
1198621198942sdot 119862
1198962+ 3 sum
0le119896119895le119894
1198621198944)
(54)
6 ISRN Probability and Statistics
The terms obtained in (54) may further be bounded fromabove in the following way
sum
0le119894119896le119895
1198621198942sdot 119862
1198962=
119899minus1
sum
119895=0
119895
sum
119894119896=0
1198621198942119862
1198962
le 119899
119899minus1
sum
119894119896=0
1198621198942119862
1198962= 119899(
119899minus1
sum
119894=0
1198621198942)
2
sum
0le119896119895le119894
1198621198944
=
119899minus1
sum
119894=0
119894
sum
119895119896=0
1198621198944
le
119899minus1
sum
119894=0
(119894 + 1) (119894 + 1) 1198621198944
=
119899minus1
sum
119894=0
(119894 + 1)2119862
1198944
(55)
We thus get the inequality
1198641198784
119899le 4([119899
119899minus1
sum
119894=0
1198621198942]
2
+ 3119899
119899minus1
sum
119894=0
(119894 + 1)2119862
1198944) (56)
We shall now focus on estimating 1198621199032
and 1198621199034 Since 119880
1199051
and 1198801199052
are associated uniformly [0 1] distributed rvrsquos 1198831199051
and 1198831199052
mdashas monotone functions of these rvrsquosmdashare associat-ed as well In order to bound
1198621199032
= sup0le1199051lt1199052le119899 1199052minus1199051=119903
Cov (1198831199051
1198831199052
) (57)
let us notice that from Schwarz inequality
Cov (1198831199051
1198831199052
) = 119864 (119868 [1198801199051
le 119909] 119868 [1198801199052
le 119909]) minus 1199092
le radic1199092 minus 1199092le 119909
(58)
On the other hand invoking inequalities from [13 15]
Cov (1198831199051
1198831199052
)
le sup119909isin[01]
[119875 (1198801199051
le 1199091198801199052
le 119909) minus 119875 (1198801199051
le 119909)119875 (1198801199052
le 119909)]
le 119862 sdot Cov13
(1198801199051
1198801199052
) for associated rvrsquos4 sdot Cov (119880
1199051
1198801199052
) for MTP2rvrsquos
(59)
As a consequence
1198621199032
le min 119909 119862 sdot Cov13
(1198801199051
1198801199052
) for associated rvrsquosmin 119909 4 sdot Cov (119880
1199051
1198801199052
) for MTP2rvrsquos
(60)
where 1199052minus 119905
1= 119903 Still we need the upper bound for 119862
1199034 It
turns out that119862
1199034
= sup1003816100381610038161003816100381610038161003816100381610038161003816
Cov2
prod
119894=1
(119868 [119880119905119894
le 119909] minus 119909)
4
prod
119894=3
(119868 [119880119905119894
le 119909] minus 119909)
1003816100381610038161003816100381610038161003816100381610038161003816
(61)
In other words among all divisions of the group 119883119905119894
119894 isin
1 2 3 4 into two subgroups 1198621199034
attains the biggest valuein case we take two subgroups consisted of two rvrsquos Thesupremum runs over the set 119905
1 119905
2 119905
3 119905
4isin N 0 le 119905
1lt 119905
2lt
1199053lt 119905
4le 119899 and 119905
3minus 119905
2= 119903 Elementary calculation leads to the
following formula
1198621199034
= sup 10038161003816100381610038161003816119875 (119880119905119894
le 119909 119894 isin 1 2 3 4)
minus 119875 (119880119905119894
le 119909 119894 isin 1 2) 119875 (119880119905119894
le 119909 119894 isin 3 4)
minus 119909 [119875 (119880119905119894
le 119909 119894 isin 1 2 3)
minus 119875 (119880119905119894
le 119909 119894 isin 1 2) 119875 (1198801199053
le 119909)
+ 119875 (119880119905119894
le 119909 119894 isin 1 3 4)
minus 119875 (119880119905119894
le 119909 119894 isin 3 4) 119875 (1198801199051
le 119909)
+ 119875 (119880119905119894
le 119909 119894 isin 1 2 4)
minus 119875 (119880119905119894
le 119909 119894 isin 1 2) 119875 (1198801199054
le 119909)
+ 119875 (119880119905119894
le 119909 119894 isin 2 3 4)
minus119875 (119880119905119894
le 119909 119894 isin 3 4) 119875 (1198801199052
le 119909)]
+ 1199092[119875 (119880
1199051
le 1199091198801199053
le 119909)
minus 119875 (1198801199051
le 119909)119875 (1198801199053
le 119909)
+ 119875 (1198801199051
le 1199091198801199054
le 119909)
minus 119875 (1198801199051
le 119909)119875 (1198801199054
le 119909)
+ 119875 (1198801199052
le 1199091198801199053
le 119909)
minus 119875 (1198801199052
le 119909)119875 (1198801199053
le 119909)
+ 119875 (1198801199052
le 1199091198801199054
le 119909)
minus119875 (1198801199052
le 119909)119875 (1198801199052
le 119909)]10038161003816100381610038161003816
le10038161003816100381610038161198771
1003816100381610038161003816 +10038161003816100381610038161198772
1003816100381610038161003816 +10038161003816100381610038161198773
1003816100381610038161003816
(62)
where for the sake of simplicity 1198771 119877
2 and 119877
3are respec-
tively the free coefficient the expression with 119909 and theexpression with 119909
2 Using the Lebowitz inequality (see [15]for instance) and inequalities obtained in [13 15] we arrive at10038161003816100381610038161198771
1003816100381610038161003816 =10038161003816100381610038161003816119875 (119880
119905119894
le 119909 119894 isin 1 2 3 4)
minus119875 (119880119905119894
le 119909 119894 isin 1 2) 119875 (119880119905119894
le 119909 119894 isin 3 4)10038161003816100381610038161003816
le 11986711988011990511198801199053
+ 11986711988011990511198801199054
+ 11986711988011990521198801199053
+ 11986711988011990521198801199054
ISRN Probability and Statistics 7
le
119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
)) for associated rvrsquos
4(
2
sum
119894=1
4
sum
119895=3
Cov (119880119905119894
119880119905119895
)) for MTP2rvrsquos
(63)
Analogously we get
10038161003816100381610038161198772
1003816100381610038161003816
le
2119909 sdot 119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
)) for associated rvrsquos
2119909 sdot 4(
2
sum
119894=1
4
sum
119895=3
Cov (119880119905119894
119880119905119895
)) for MTP2rvrsquos
10038161003816100381610038161198773
1003816100381610038161003816
le
1199092sdot 119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
) ) for associated rvrsquos
1199092sdot 4(
2
sum
119894=1
4
sum
119895=3
Cov (119880119905119894
119880119905119895
)) for MTP2rvrsquos
(64)
Eventually we have
1198621199034
le
119891 (119909) sdot119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
)) for associated rvrsquos
119891 (119909) sdot4(
2
sum
119894=1
4
sum
119895=3
Cov (119880119905119894
119880119905119895
)) for MTP2rvrsquos
(65)
where 1199053minus 119905
2= 119903 and 119891(119909) = (1 + 2119909 + 119909
2) for 119909 isin [0 1]
Let us now introduce the following notation
120579119903=
sup119905isinN
Cov (119880119905 119880
119905+119903) for 119903 isin 1 2
1 for 119903 = 0
(66)
and assume it decays powerly at rate 119886 in the following way
120579119903= 119874(
1
(119903 + 1)119886) for 119886 gt 0 119903 isin 0 1 (67)
Let us get back to inequality (56) we can now carry on Atfirst for associated rvrsquos
1198641198784
119899le 4[119899
119899minus1
sum
119903=0
min 119909 119862 sdot Cov13(119880
1199051
1198801199052
)]
2
+ 4 sdot 3119899
119899minus1
sum
119903=0
(119903 + 1)2119891 (119909) sdot 119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
))
le 119863 sdot ([119899
119899minus1
sum
119903=0
min 119909 (119903 + 1)minus1198863
]
2
+119899
119899minus1
sum
119903=0
(119903 + 1)2(119903 + 1)
minus1198863)
= 119863 sdot ([119899
119899
sum
119903=1
min 119909 119903minus1198863]
2
+ 119899
119899
sum
119903=1
1199032119903minus1198863
)
le 119863 sdot ([
[
119899 sum
119903lt119909minus3119886
119909 + 119899 sum
119903ge119909minus3119886
1
1199031198863]
]
2
+ 119899
119899
sum
119903=1
1199032minus1198863
)
le 1198631sdot (119899
2119909
2(119886minus3)119886+ 120585)
(68)
where119863 and1198631are constants and
120585 = 119899
119899
sum
119903=1
1199032minus1198863
=
119874(119899) 119886 gt 9
119874 (119899 ln 119899) 119886 = 9
119874 (1198994minus1198863
) 3 lt 119886 lt 9
(69)
It is worth mentioning that in the last inequality of (68) weused the estimate
int
infin
119909
1
119905119901119889119905 sim
1
119909119901minus1for 119901 gt 1 (70)
At the same time in the case of MTP2rvrsquos we get
1198641198784
119899le 119863
2sdot (119899
2119909
2(119886minus1)119886+ 120577) (71)
where1198632is constant and
120577 = 119899
119899
sum
119903=1
1199032minus119886
=
119874(119899) 119886 gt 3
119874 (119899 ln 119899) 119886 = 3
119874 (1198994minus119886
) 1 lt 119886 lt 3
(72)
Let now 120572119899(119909) = (1radic119899)sum
119899
119894=1(119868[119880
119905119894
le 119909] minus 119909) Then
120572119899(119909) minus 120572
119899(119910) =
1
radic119899
119899
sum
119894=1
(119868 [119909 lt 119880119905119894
le 119910] minus (119909 minus 119910))
for 119909 119910 isin [0 1]
(73)
8 ISRN Probability and Statistics
has the fourth moment estimatedmdashin the case of associatedrvrsquosmdashby
119864[120572119899(119909) minus 120572
119899(119910)]
4
= 119864[1
radic119899
119899
sum
119894=1
(119868 [119909 lt 119880119905119894
le 119910] minus (119909 minus 119910))]
4
=1
1198992119864[
119899
sum
119894=1
(119868 [119909 lt 119880119905119894
le 119910] minus (119909 minus 119910))]
4
le 1198631sdot (
1
119899211989921003816100381610038161003816119909 minus 119910
1003816100381610038161003816
2(119886minus3)119886
+120585
1198992)
= 1198631sdot (
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2(119886minus3)119886
+120585
1198992)
=
119874(119899minus1) 119886 gt 9
119874(ln 119899119899
) 119886 = 9
119874 (1198992minus1198863
) 3 lt 119886 lt 9
(74)
and in the case of MTP2rvrsquos by
119864[120572119899(119909) minus 120572
119899(119910)]
4
le 1198632sdot (
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2(119886minus1)119886
+120577
1198992)
=
119874(119899minus1) 119886 gt 3
119874(ln 119899119899
) 119886 = 3
119874 (1198992minus119886
) 1 lt 119886 lt 3
(75)
In light of the Shao and Yursquos criterion our process is tight forassociated rvrsquos when 119886 gt 6 and for MTP
2rvrsquos when 119886 gt 2
Let us sum up this result in the following theorem
Theorem 3 Let 120572119899(119909) = (1radic119899)sum
119899
119894=1(119868[119880
119894le 119909] minus 119909) be the
empirical process built on an associated sequence of uniformly[0 1] distributed rvrsquos 119880
119894119894ge1
Let also
120579119903= sup
119905isinNCov (119880
119905 119880
119905+119903) = 119874 ((119903 + 1)
minus119886)
where 119903 isin 0 1 119886 gt 0
(76)
Then 120572119899(119909)
119899ge1is tight for 119886 gt 6 If the rvrsquos 119880
119894119894ge1
are MTP2
then the process is tight for 119886 gt 2
Yu assumed stationarity of 119880119894119894ge1
andsum
infin
119899=111989965+]Cov(119880
0 119880
119899) lt infin for a positive constant ]
thus the rate of decay 119886 gt 7 5 Our result weakens consid-erably these assumptions especially in the case of MTP
2rvrsquos
Louhichi in [16] proposed a different tightness criterioninvolving the so-called bracketing numbers She managedto enhance Yursquos resultmdasheven more than Shao and Yu in[8]mdashsince she proved that it suffices to take 119886 gt 4 to gettightness of the empirical process based on the associated
rvrsquos Nevertheless she kept the assumption of stationarityvalid
In the final analysis our resultrsquos advantage is the absenceof the stationarity assumption and the rate of decay for120579119903remains (up to the authorrsquos knowledge) unimproved for
MTP2rvrsquos
Unfortunately with a view to obtainingweak convergenceof the process in question that is also convergence offinite-dimensional distributions we do not know how tomanage without the assumption of stationarityTherefore weconclude with the following corollary
Corollary 4 Let 120572119899(119909) = (1radic119899)sum
119899
119894=1(119868[119880
119894le 119909] minus 119909) be the
empirical process built on a stationary associated sequence ofuniformly [0 1] distributed rvrsquos 119880
119894119894ge1
Let also
120579119903= Cov (119880
1 119880
1+119903) = 119874 ((119903 + 1)
minus119886)
where 119903 isin 0 1 119886 gt 0
(77)
Then if 119886 gt 6
120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (78)
where 119861(sdot) is the zero mean Gaussian process on [0 1] withcovariance structure defined by
1205902(119909 119910) = 119909 and 119910 minus 119909119910
+
infin
sum
119895=1
[ Cov (119868 [1198801le 119909] 119868 [119880
119895+1le 119910])
+ Cov (119868 [1198801le 119910] 119868 [119880
119895+1le 119909])]
(79)
If the rvrsquos 119880119894119894ge1
are MTP2 then it suffices to be 119886 gt 2 in
order to claim the above convergence
Proof It remains to establish convergence of finite-dimen-sional distributions repeating the procedure from [7]
4 Tightness of the Kernel-TypeEmpirical Process
In this section we shall weaken assumption imposed on thecovariance structure of rvrsquos 119883
119895119895ge1
by Cai and Roussas in [1]for the kernel estimator of the df
119865119899(119909) =
1
119899
119899
sum
119895=1
119870(
119909 minus 119883119895
ℎ119899
) (80)
They deal with a stationary sequence of negatively associatedrvrsquos (cf [17]) and need the same condition as Yu [7] that is
1003816100381610038161003816Cov (1198831 119883
1+119899)1003816100381610038161003816 = 119874(
1
11989975+]) (81)
to get tightness of the smooth empirical process (see condi-tion (A4) in [1])
ISRN Probability and Statistics 9
It turns out that it suffices to have
1003816100381610038161003816Cov (1198801 119880
1+119899)1003816100381610038161003816 = 119874(
1
1198993119901(119901minus1)) (82)
where 119901 gt 4 is a positive constant taken from the tightnesscriterion (8) It is easy to see that asymptotically we get therate 3
On the way to prove it we will also take use of aRosenthal-type inequality due to Shao andYu (seeTheorem 2in [8]) we shall recall in the following lemma
Lemma 5 Let 119901 gt 2 and 119891 be a real valued function boundedby 1 with bounded first derivative Suppose that 119883
119899119899ge1
is asequence of stationary and associated rvrsquos such that for 119899 isin N
Cov (1198831 119883
119899) = 119874 (119899
minus119887) for some 119887 gt 119901 minus 1 (83)
Then for any 120583 gt 0 there exists some positive constant 119896120583
independent of the function 119891 for which
1198641003816100381610038161003816119878119899
(119891) minus 119864119878119899(119891)
1003816100381610038161003816
119901
le 119896120583(119899
1+12058310038171003817100381710038171003817119891
101584010038171003817100381710038171003817
2
+1198991199012 [
[
119899
sum
119895=1
10038161003816100381610038161003816Cov (119891 (119883
1)119891 (119883
119895))10038161003816100381610038161003816]
]
)
(84)
As we can see the lemma assumes association but itworks for negatively associated rvrsquos as well since in the proofit reaches back the result of Newman (see Proposition 15 in[18]) where both types of association are allowed
Let us recall that 120572119899(119909) = radic119899(119865
119899(119909) minus 119864119865
119899(119909)) where
119865119899(119909) = (1119899)sum
119899
119895=1119870((119909 minus 119883
119895)ℎ
119899) It is easy to see that
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
= 119899minus1199012
11986410038161003816100381610038161003816119878119899(119891 (119883
119895)) minus 119864119878
119899(119891 (119883
119895))10038161003816100381610038161003816
119901
(85)
where
119891 (119883119895) = 119870(
119909 minus 119883119895
ℎ119899
) minus 119870(
119910 minus 119883119895
ℎ119899
)
119878119899(119891) =
119899
sum
119895=1
119891 (119883119895)
(86)
With an intent to use Lemma 5 we need 119891 to be boundedfrom above by 1 (which in our case is obvious) and to havebounded 119891
1015840 thus we assume
sup119905isinR
10038161003816100381610038161003816119870
1015840(119905)
10038161003816100381610038161003816
1
ℎ119899
le119862
119870
ℎ119899
lt infin (87)
Now applying inequality (84) we have
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
le119896120583119899minus1199012
1198991+120583
1198622
119870
ℎ2
119899
+1198991199012[
[
119899
sum
119895=1
10038161003816100381610038161003816Cov (119891 (119883
1)119891 (119883
119895))10038161003816100381610038161003816]
]
1199012
(88)
Newman showed in [18] that
Cov (1198911(119883) 119891
2(119884))
= ∬R2119891
1015840
1(119909) 119891
1015840
2(119910)
times [119875 (119883 le 119909 119884 le 119910) minus 119875 (119883 le 119909) 119875 (119884 le 119910)] 119889119909 119889119910
(89)
if 1198911and 119891
2are real valued functions on R having square
integrable derivatives 1198911015840
1and 119891
1015840
2 respectively and provided
that 1198911(119883) 119891
2(119884) have finite secondmoments In light of that
equation and linearity of covariance we get10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
=
1003816100381610038161003816100381610038161003816∬
R2[119875 (119883
1lt 119909 minus ℎ
119899119904 119883
119895lt 119909 minus ℎ
119899119905)
minus119875 (1198831lt 119909 minus ℎ
119899119904 119883
119895lt 119910 minus ℎ
119899119905)]
minus [119875 (1198831lt 119909 minus ℎ
119899119904) 119875 (119883
119895lt 119909 minus ℎ
119899119905)
minus119875 (1198831lt 119909 minus ℎ
119899119904) 119875 (119883
119895lt 119910 minus ℎ
119899119905)]
minus [119875 (1198831lt 119910 minus ℎ
119899119904 119883
119895lt 119909 minus ℎ
119899119905)
minus119875 (1198831lt 119910 minus ℎ
119899119904 119883
119895lt 119910 minus ℎ
119899119905)]
minus [119875 (1198831lt 119910 minus ℎ
119899119904) 119875 (119883
119895lt 119910 minus ℎ
119899119905)
minus 119875 (1198831lt 119910 minus ℎ
119899119904)
times 119875 (119883119895lt 119909 minus ℎ
119899119905)] 119896 (119904) 119896 (119905) 119889119904 119889119905
10038161003816100381610038161003816100381610038161003816
(90)
Without the loss of generality we may and do assume that119909 gt 119910 thus10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
=
1003816100381610038161003816100381610038161003816∬
R2119875 (119910 minus ℎ
119899119904 le 119883
1lt 119909 minus ℎ
119899119904
119910 minus ℎ119899119905 le 119883
119895lt 119909 minus ℎ
119899119905)
minus 119875 (119910 minus ℎ119899119904 le 119883
1lt 119909 minus ℎ
119899119904)
times 119875 (119910 minus ℎ119899119905 le 119883
119895lt 119909 minus ℎ
119899119905) 119896 (119904) 119896 (119905) 119889119904 119889119905
1003816100381610038161003816100381610038161003816
(91)
and by triangle inequality we get10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
le ∬R2[int
119909minusℎ119899119904
119910minusℎ119899119904
int
119909minusℎ119899119905
119910minusℎ119899119905
1198911198831119883119895(119906 V) 119889119906 119889V
+int
119909minusℎ119899119904
119910minusℎ119899119904
1198911198831(119906) 119889119906int
119909minusℎ119899119905
119910minusℎ119899119905
119891119883119895(V) 119889V] 119896 (119904) 119896 (119905) 119889119904 119889119905
(92)
10 ISRN Probability and Statistics
1198911198831119883119895
(119906 V) 1198911198831
(119906) and 119891119883119895
(V) are the joint pdf of [1198831 119883
119895]
and marginal pdfrsquos of rvrsquos1198831and119883
119895 respectively We need
to further assume that 1198621stands for a common upper bound
of 1198911198831
and 1198911198831119883119895
for all 119895 isin N that is
100381710038171003817100381710038171198911198831
10038171003817100381710038171003817le 119862
1
1003817100381710038171003817100381710038171198911198831119883119895
100381710038171003817100381710038171003817le 119862
1forall119895 ge 1 (93)
Then we finally obtain
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816le 2119862
1
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
where = max 1198621 119862
2
1
(94)
On the other hand proceeding like Cai and Roussas in [1] wecan shortly get
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
le
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119909 minus 119883
1
ℎ119899
) 119870(
119909 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119909 minus 119883
1
ℎ119899
) 119870(
119910 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119910 minus 119883
1
ℎ119899
) 119870(
119909 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119910 minus 119883
1
ℎ119899
) 119870(
119910 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
le 41198622
10038161003816100381610038161003816Cov (119883
1 119883
119895)10038161003816100381610038161003816
13
(95)
where 1198622is a constant relevant to the covariance inequality
for negatively associated rvrsquos (see [13])We now arrive at the following inequality
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
le max 1198621 119862
2
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
10038161003816100381610038161003816Cov(119883
1 119883
119895)10038161003816100381610038161003816
13
(96)
Assuming |Cov(119891(1198831) 119891(119883
119895))| = 119874(119895
minus119903) and proceeding
similarly to (68) we can get
[
[
119899
sum
119895=1
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816]
]
1199012
le max 1198621 119862
2 [
[
sum
119895lt|119909minus119910|minus2(1199033)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
+ sum
119895ge|119909minus119910|minus2(1199033)
1
1198951199033]
]
1199012
le 2max 1198621 119862
21003816100381610038161003816119909 minus 119910
1003816100381610038161003816
((119903minus3)119903)119901
(97)
To sum up we obtain the following inequality
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
le 119896120583119899
1+120583minus1199012119862
2
119870
ℎ2
119899
+ 2max 1198621 119862
21003816100381610038161003816119909 minus 119910
1003816100381610038161003816
((119903minus3)119903)119901
le 1198621198991+120583minus1199012 1
ℎ2
119899
+1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
((119903minus3)119903)119901
(98)
where 119862 = 119896120583max1198622
119896 2max119862
1 119862
2 The formula of the
tightness criterion (8) implies that ((119903 minus 3)119903)119901 gt 1 so
119903 gt3119901
119901 minus 1 where 119901 gt 2 (99)
Let us recall the assumptions imposed on the bandwidhtsℎ
119899119899ge1
by Cai and Roussas in [1]B1 lim
119899rarrinfinℎ119899= 0 and ℎ
119899gt 0 for all 119899 isin N
+
B2 lim119899rarrinfin
119899ℎ119899= infin thus ℎ
119899= 119874(1119899
1minus120573) 120573 gt 0
B3 lim119899rarrinfin
119899ℎ4
119899= 0 hence ℎ
119899= 119874(1119899
14+120575) 120575 gt 0
In light of the above
1198991+120583minus1199012 1
ℎ2
119899
= 11989932+120583+2120575minus1199012
(100)
where 120583 gt 0 119901 gt 2 and 0 lt 120575 lt 34 Confrontation with thetightness criterion (8) forces
3
2+ 120583 + 2120575 minus
119901
2lt minus
1
2 (101)
which implies 119901 gt 4 Let us conclude with the followingtheorem
Theorem6 Let 120572119899(119909) = radic119899(119865
119899(119909)minus119864119865
119899(119909)) be the empirical
process built on the kernel estimator of the df119865 for a stationarysequence of negatively associated rvrsquos Assume that conditionsA1 A2 (87) B1 B2 B3 are satisfied Then provided that thetightness criterion (8) holds with 119901 gt 4 it suffices to demand
1003816100381610038161003816Cov (1198801 119880
1+119899)1003816100381610038161003816 = 119874(
1
1198993119901(119901minus1)) (102)
in order to obtain120572
119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (103)
where 119861(sdot) is the zero mean Gaussian process with covariancestructure defined by
1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)
+
infin
sum
119895=1
[ Cov (119868 [1198831le 119909] 119868 [119883
119895+1le 119910])
+ Cov (119868 [1198831le 119910] 119868 [119883
119895+1le 119909])]
(104)
and 119909 119910 isin [0 1]
Proof The remaining covergence of finite-dimensional dis-tributions of 120572
119899(119909) is established in [1]
ISRN Probability and Statistics 11
5 Weak Convergence of theRecursive Kernel-Type Empirical Processunder IID Assumption
The aim of this section is to show weak convergence of theempirical process
120572119899(119909) =
1
radic119899
119899
sum
119895=1
[119870(
119909 minus 119883119895
ℎ119895
) minus 119864119870(
119909 minus 119883119895
ℎ119895
)] (105)
built on iid rvrsquos 119883119895119895ge1
Wewill prove that120572
119899(sdot) convergesweakly to the Brownian
bridge 119861(sdot) with the following covariance structure
1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910) for 119909 119910 isin [0 1]
(106)
It turns out that the tightness criterion (8) suffices to reachthe goal Convergence of finite-dimensional distributions of120572
119899(sdot) holds and we shall show itProceeding like Cai and Roussas in [1] we need to show
that for any 119886 119887 isin R
119886120572119899(119909) + 119887120572
119899(119910) 997888rarr 119886119861 (119909) + 119887119861 (119910) in distribution
(107)
Let us introduce the notation
119884119894(119909) = 119870(
119909 minus 119883119894
ℎ119894
) minus 119864119870(119909 minus 119883
119894
ℎ119894
) (108)
and look into the covariance structure of 120572119899(sdot)
Cov (120572119899(119909) 120572
119899(119910))
= Cov( 1
radic119899
119899
sum
119894=1
119884119894(119909)
1
radic119899
119899
sum
119894=1
119884119894(119910))
=1
119899
119899
sum
119894=1
Cov (119884119894(119909) 119884
119894(119910))
=1
119899
119899
sum
119894=1
Cov(119870(119909 minus 119883
119894
ℎ119894
) 119870(119910 minus 119883
119894
ℎ119894
))
(109)
where the second equality follows from assumed indepen-dence of rvrsquos 119883
119894119894ge1
Firstly we observe that each summand converges to119865(119909and
119910) minus 119865(119909)119865(119910) since
Cov(119870(119909 minus 119883
119894
ℎ119894
) 119870(119910 minus 119883
119894
ℎ119894
))
= 119864[119870(119909 minus 119883
119894
ℎ119894
)119870(119910 minus 119883
119894
ℎ119894
)]
minus 119864119870(119909 minus 119883
119894
ℎ119894
)119864119870(119910 minus 119883
119894
ℎ119894
)
(110)
where
119864[119870(119909 minus 119883
119894
ℎ119894
)119870(119910 minus 119883
119894
ℎ119894
)]
= int
119909and119910
minusinfin
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
+ int
119909or119910
119909and119910
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
+ int
infin
119909or119910
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
(111)
and 119865 denotes the common df of the rvrsquos 119883119894119894ge1
In [6] itwas shown that
119864119870(119909 minus 119883
119894
ℎ119894
) 997888rarr 119865 (119909) (112)
and recalling that the kernel function 119870 is a df as well wearrive at the conclusion
Secondly applying Toeplitz lemma we get
Cov (120572119899(119909) 120572
119899(119910)) 997888rarr 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)
= 1205902(119909 119910) 119899 997888rarr infin
(113)
Thus
Var (119886120572119899(119909) + 119887120572
119899(119910))
= 1198862 Var (120572
119899(119909)) + 2119886119887Cov (120572
119899(119909) 120572
119899(119910))
+ 1198872 Var (120572
119899(119910))
997888rarr 1198862120590
2(119909 119909) + 2119886119887120590
2(119909 119910) + 119887
2120590
2(119910 119910) 119899 997888rarr infin
(114)
We are now on the way to prove that 119886120572119899(119909) + 119887120572
119899(119910) con-
verges in distribution to 119886119861(119909) + 119887119861(119910) sim N(0 1198862120590
2(119909 119909) +
21198861198871205902(119909 119910) + 119887
2120590
2(119910 119910)) which in other words means that
1
radic119899
infin
sum
119894=1
119886119884119894(119909) + 119887119884
119894(119910)
radicVar (119886120572119899(119909) + 119887120572
119899(119910))
997888rarr N (0 1) (115)
In order to obtain (115) we will use Lyapunov condition
lim119899rarrinfin
1
1198632+120575
119899
119899
sum
119894=1
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884119894(119910)
1003816100381610038161003816
2+120575
= 0 (116)
for 120575 = 1 where 1198632
119899= Var(sum119899
119894=1[119886119884
119894(119909) + 119887119884
119894(119910)]) Since
|119870(sdot) minus 119864119870(sdot)| le 2
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884i (119910)1003816100381610038161003816
3
le 8 (1198863+ 119887
3) (117)
Moreover 1198632
119899= 119899Var(119886120572
119899(119909) + 119887120572
119899(119910)) together with (114)
yields
1198633
119899= 119874 (119899
32) (118)
12 ISRN Probability and Statistics
Combining (117) with (118) we obtain
1
1198633
119899
119899
sum
119894=1
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884119894(119910)
1003816100381610038161003816
3
= 119874 (119899minus12
) (119)
which shows that Lyapunov condition holds and completesthe proof of convergence of finite-dimensional distributionsof 120572
119899(119909)We summarize the result of that section in the following
theorem
Theorem 7 Let 120572119899(119909) be the recursive kernel-type empirical
process defined by (105) built on iid rvrsquos 119883119895119895ge1
withmarginal df 119865 If 120572
119899(sdot) satisfies the tightness criterion (8) then
120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (120)
where 119861 is the Brownian bridge
Acknowledgment
The author would like to thank the referees for the carefulreading of the paper and helpful remarks
References
[1] Z Cai and G G Roussas ldquoWeak convergence for smoothestimator of a distribution function under negative associationrdquoStochastic Analysis and Applications vol 17 no 2 pp 145ndash1681999
[2] Z Cai andG G Roussas ldquoEfficient Estimation of a DistributionFunction under Qudrant Dependencerdquo Scandinavian Journal ofStatistics vol 25 no 1 pp 211ndash224 1998
[3] ZW Cai and G G Roussas ldquoUniform strong estimation under120572-mixing with ratesrdquo Statistics amp Probability Letters vol 15 no1 pp 47ndash55 1992
[4] G G Roussas ldquoKernel estimates under association stronguniform consistencyrdquo Statistics amp Probability Letters vol 12 no5 pp 393ndash403 1991
[5] Y F Li and S C Yang ldquoUniformly asymptotic normalityof the smooth estimation of the distribution function underassociated samplesrdquo Acta Mathematicae Applicatae Sinica vol28 no 4 pp 639ndash651 2005
[6] Y Li C Wei and S Yang ldquoThe recursive kernel distributionfunction estimator based onnegatively and positively associatedsequencesrdquo Communications in Statistics vol 39 no 20 pp3585ndash3595 2010
[7] H Yu ldquoA Glivenko-Cantelli lemma and weak convergence forempirical processes of associated sequencesrdquo ProbabilityTheoryand Related Fields vol 95 no 3 pp 357ndash370 1993
[8] Q M Shao and H Yu ldquoWeak convergence for weightedempirical processes of dependent sequencesrdquo The Annals ofProbability vol 24 no 4 pp 2098ndash2127 1996
[9] P Billingsley Convergence of Probability Measures Wiley Seriesin Probability and Statistics Probability and Statistics JohnWiley amp Sons New York NY USA 1999
[10] J D Esary F Proschan and D W Walkup ldquoAssociation ofrandom variables with applicationsrdquo Annals of MathematicalStatistics vol 38 pp 1466ndash1474 1967
[11] A Bulinski and A Shashkin Limit Theorems for AssociatedRandom Fields and Related Systems vol 10 of Advanced Serieson Statistical Science and Applied Probability World Scientific2007
[12] S Karlin and Y Rinott ldquoClasses of orderings of measures andrelated correlation inequalities I Multivariate totally positivedistributionsrdquo Journal of Multivariate Analysis vol 10 no 4 pp467ndash498 1980
[13] P Matuła and M Ziemba ldquoGeneralized covariance inequali-tiesrdquo Central European Journal of Mathematics vol 9 no 2 pp281ndash293 2011
[14] P Doukhan and S Louhichi ldquoA new weak dependence condi-tion and applications to moment inequalitiesrdquo Stochastic Proc-esses and their Applications vol 84 no 2 pp 313ndash342 1999
[15] P Matuła ldquoA note on some inequalities for certain classes ofpositively dependent random variablesrdquo Probability and Math-ematical Statistics vol 24 no 1 pp 17ndash26 2004
[16] S Louhichi ldquoWeak convergence for empirical processes of asso-ciated sequencesrdquo Annales de lrsquoInstitut Henri Poincare vol 36no 5 pp 547ndash567 2000
[17] K Joag-Dev and F Proschan ldquoNegative association of randomvariables with applicationsrdquoThe Annals of Statistics vol 11 no1 pp 286ndash295 1983
[18] C M Newman ldquoAsymptotic independence and limit theoremsfor positively and negatively dependent random variablesrdquo inInequalities in Statistics and Probability Y L Tong Ed vol 5pp 127ndash140 Institute ofMathematical StatisticsHayward CalifUSA 1984
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
ISRN Probability and Statistics 3
where
119865119899(119909) =
1
119899
119899
sum
119895=1
119870(
119909 minus 119883119895
ℎ119899119895
) (16)
is the generalized kernel estimator of the df119865 ApplyingChe-byshevrsquos inequality we shall find 119886 satisfying
119886 geradic119899
120578sdot 119864
10038161003816100381610038161003816119865119899(119909) minus 119864119865
119899(119909)
10038161003816100381610038161003816forall119899 isin N (17)
Such 119886 exists if 119864|120572119899(119909)| lt infin for all 119899 ge 1 Implementing
119910 = 0 and 1199013= 0 in the assumption (8) we get for 119901 gt 2
1199011 119901
2gt 1
1198641003816100381610038161003816120572119899
(119909)1003816100381610038161003816
119901
le 119862 sdot(|119909|1199011+119899
minus11990122|119909|
1199013) le 119862 sdot(1+
1
11989911990122) le 2119862
(18)
Now applying Holderrsquos inequality we arrive at
1198641003816100381610038161003816120572119899
(119909)1003816100381610038161003816 le
4radic119864
1003816100381610038161003816120572119899(119909)
1003816100381610038161003816
4
(19)
which in light of the inequality (18) for 119901 = 4 is boundedfrom above by 4radic2119862 lt infin Therefore condition (9) holds
We shall now proceed to checking condition (10) Let usrecall that the modulus of continuity of the function 119909 isin
119862[0 1] 120575 gt 0 is given by the formula
1199081(119909 120575) = sup
0le119910le1minus120575
sup119910le119909le119910+120575
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 (20)
As it is shown in [9] (page 123) there is a relation between119908
1(sdot sdot) and 1199081015840
1(sdot sdot) in the spaces 119862[0 1] and119863[0 1] relatively
Namely for 119909 isin 119863[0 1] and 120575 lt 12
1199081(119909 2120575) ge 119908
1015840
1(119909 120575) (21)
Thus
119875 (1199081015840
1(120572
119899 120575) ge 120598) le 119875 (119908
1(120572
119899 2120575) ge 120598) (22)
Therefore condition (10) holds if we show
forall120598gt0
forall120578gt0
exist0le120575lt12
exist1198990
forall119899ge1198990
119875 (1199081(120572
119899 2120575) ge 120598) le 120578 (23)
With a view to obtaining the desired inequality we shall pro-ceed patiently in five steps
Step 1 We need a moment inequality for the rv 120572119899(119909)minus120572
119899(119910)
involving the distance between the points 119909 and 119910 Let usthen assume that for the constants 119862 119901 119901
1 119901
2 119901
3given in
Theorem 1 inequality (8) holds Fix 120598 gt 0 120578 gt 0 and definethe quantity 119903
119899= 120598radic119899 119899 isin N Next we fix 119909 119910 isin [0 1] and
take 119899 large enough so that 119903119899le |119909minus119910|Then 119899minus12
le |119909minus119910|120598minus1
and we have
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
le 119862 (1 + 120598minus1199012)1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
min11990111199012+1199013
(24)
Step 2 Let us now fix 119899 isin N 120575 isin (0 1) and consider the fol-lowing rvrsquos
120594119894= 120572
119899(119910 + 119894119903
119899) minus 120572
119899(119910 + (119894 minus 1) 119903
119899) (25)
for 119894 isin 1 2 119898119899 where119898
119899= 119898(119899 120575) is such that 119903
119899119898
119899le 120575
It is easy to see that for 119878119894= sum
119894
119896=1120594
119896
max1le119894le119898
119899
1003816100381610038161003816119878119894
1003816100381610038161003816 = max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119903
119899) minus 120572
119899(119910)
1003816100381610038161003816 (26)
Let us notice that for rvrsquos 120594119894119894ge1
the conditions of Theo-rem 102 in [9] are satisfied with 4120573 = 119901 119906
119897= 119903
119899forall
119894lt119897le119895and
2120572 = min1199011 119901
2+ 119901
3 Therefore we are equipped with the
following maximal inequality
119875( max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119903
119899) minus 120572
119899(119910)
1003816100381610038161003816 ge 120582)
le
119862119901min119901
11199012+1199013
120582119901(119898
119899119903119899)min119901
11199012+1199013
(27)
for all 120582 gt 0
Step 3 Let 119872 = sup119909isin[01]
119891(119909) where 119891 is the pdf of rvrsquos119883
119895119895ge1
For fixed 120598 120578 gt 0 let us take 120575 gt 0 such that
119862119901min119901
11199012+1199013
(119872120598)119901
sdot (2120575)min119901
11199012+1199013lt 120578 (28)
and define119898119899= lfloor120575119903
119899rfloor For sufficiently large 119899 isin N we have
119898119899ge 1 and then we get
119903119899119898
119899le 120575 lt (119898
119899+ 1) 119903
119899le 2119898
119899119903119899le 120575 (29)
Step 4 Our goal is to obtain an inequality which enables usto bound the supremum of the increment of the function 120572
119899
via the maximum of the increments of that function on somesubintervals To bemore precise wewill find the upper boundfor
sup119910le119909le119910+119898
119899119902
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 (30)
in terms of
max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572
119899(119910)
1003816100381610038161003816 (31)
where 119909 119910 isin [0 1] 119910 le 119909 le 119910 + 119902 119902 ge 120598119899 and119898119899is defined
as in Step 3Let us recall that 120572
119899(119909) = radic119899(119865
119899(119909) minus 119864119865
119899(119909)) where
119865119899(119909) = (1119899)sum
119899
119895=1119870((119909 minus 119883
119895)ℎ
119899119895) From the triangle ine-
quality we can see that
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816
leradic119899
119899
10038161003816100381610038161003816100381610038161003816100381610038161003816
119899
sum
119895=1
[119870(
119909 minus 119883119895
ℎ119899119895
) minus 119870(
119910 minus 119883119895
ℎ119899119895
)]
10038161003816100381610038161003816100381610038161003816100381610038161003816
+ radic119899 (10038161003816100381610038161003816119864119865
119899(119909) minus 119865 (119909)
10038161003816100381610038161003816+10038161003816100381610038161003816119864119865
119899(119910) minus 119865 (119910)
10038161003816100381610038161003816
+1003816100381610038161003816119865 (119909) minus 119865 (119910)
1003816100381610038161003816 )
(32)
4 ISRN Probability and Statistics
Since 119870 is a nondecreasing function and applying triangleinequality again we get
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816
le1003816100381610038161003816120572119899
(119910 + 119902) minus 120572119899(119910)
1003816100381610038161003816 +radic119899
10038161003816100381610038161003816119864119865
119899(119910 + 119902) minus 119864119865
119899(119910)
10038161003816100381610038161003816
+ radic119899 (10038161003816100381610038161003816119864119865
119899(119909) minus 119865 (119909)
10038161003816100381610038161003816+10038161003816100381610038161003816119864119865
119899(119910) minus 119865 (119910)
10038161003816100381610038161003816
+1003816100381610038161003816119865 (119909) minus 119865 (119910)
1003816100381610038161003816 )
(33)
Cai and Roussas in [1] showed that under assumptions madeon the df 119865 and the kernel function119870 we have
10038161003816100381610038161003816119864119865
119899(119909) minus 119865 (119909)
10038161003816100381610038161003816= 119874 (ℎ
2
119899) (34)
Similarly by Taylor expansion it is easy to see that
1003816100381610038161003816119865 (119909) minus 119865 (119910)1003816100381610038161003816 le
10038171003817100381710038171198911003817100381710038171003817 sdot
1003816100381610038161003816119909 minus 1199101003816100381610038161003816 (35)
Thus
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 le1003816100381610038161003816120572119899
(119910 + 119902) minus 120572119899(119910)
1003816100381610038161003816
+ 119862radic119899ℎ4
119899+119872radic119899119902
(36)
where 119872 = sup119909isin[01]
119891(119909) and 119862 = 1198621198911015840119870
is a positive con-stant dependent on the functions 119865 and119870
Let us recall that 119910 le 119909 le 119910+ 119902119898119899= lfloor120575119903
119899rfloor and 119902 ge 120598119899
Taking 119902 = (120598radic119899)(= 119903119899) we notice that 119898
119899119902 = lfloor120575119903
119899rfloor119903
119899le 120575
thus when 119899 rarr infin the interval [119910 119910 + 119898119899119902] coincides with
[119910 119910 + 120575] Approaching the aim of Step 4 let us observe that
sup119910le119909le119910+119898
119899119902
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 = max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572
119899(119910)
1003816100381610038161003816
+1003816100381610038161003816120572119899
(1199090) minus 120572
119899(119910 + 119894max119902)
1003816100381610038161003816
(37)
where 1199090isin [119910+119894
0119902 119910+(119894
0+1)119902] is a point at which the above
supremum is attained and
119894max =
1198940+ 1 if 1003816100381610038161003816120572119899
(119910 + [1198940+ 1] 119902) minus 120572
119899(119910)
1003816100381610038161003816
ge1003816100381610038161003816120572119899
(119910 + 1198940119902) minus 120572
119899(119910)
1003816100381610038161003816
1198940
otherwise(38)
Without the loss of generality we may assume that 119894max = 1198940+
1 which implies
sup119910le119909le119910+119898
119899119902
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 = max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572
119899(119910)
1003816100381610038161003816
+1003816100381610038161003816120572119899
(1199090) minus 120572
119899(119910 + 119894
0119902)1003816100381610038161003816
(39)
Now applying inequality (36) the definition of 119894max and thetriangle inequality we have
sup119910le119909le119910+119898
119899119902
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816
le max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572
119899(119910)
1003816100381610038161003816
+1003816100381610038161003816120572119899
(119910 + [1198940+ 1] 119902) minus 120572
119899(119910 + 119894
0119902)1003816100381610038161003816
+ 119872radic119899119902 + 119862radic119899ℎ4
119899
le max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572
119899(119910)
1003816100381610038161003816
+1003816100381610038161003816120572119899
(119910 + [1198940+ 1] 119902) minus 120572
119899(119910)
1003816100381610038161003816
+1003816100381610038161003816120572119899
(119910 + 1198940119902) minus 120572
119899(119910)
1003816100381610038161003816 + 119872radic119899119902 + 119862radic119899ℎ4
119899
le 3 max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572
119899(119910)
1003816100381610038161003816 + 119872radic119899119902 + 119862radic119899ℎ4
119899
(40)
If we plug in 119902 = 119903119899we arrive at
sup119910le119909le119910+119898
119899119903119899
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 le 3 max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119903
119899) minus 120572
119899(119910)
1003816100381610038161003816
+ 119872radic119899119903119899+ 119862radic119899ℎ4
119899
(41)
Step 5 Finally we are in a position to obtain inequality (23)that is
forall120598gt0forall
120578gt0exist
0le120575lt12exist
1198990
forall119899ge1198990
119875( sup0le119910le1minus2120575
sup119910le119909le119910+2120575
1003816100381610038161003816120572119899(119909)minus120572
119899(119910)
1003816100381610038161003816
ge 120598) le 120578
(42)
We now successivelymake use the inequalities (29) (41) (27)(29) and (28) to get
119875( sup119910le119909le119910+2120575
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 ge 4119872120598 + 119862radic119899ℎ4
119899)
le 119875( sup119910le119909le119910+2(119898
119899+1)119903119899
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 ge 4119872120598 + 119862radic119899ℎ4
119899)
le 119875(3 max1le119894le2(119898119899+1)
1003816100381610038161003816120572119899(119910 + 119894119903
119899) minus 120572
119899(119910)
1003816100381610038161003816
+119872radic119899119903119899+ 119862radic119899ℎ4
119899ge 4119872120598 + 119862radic119899ℎ4
119899)
ISRN Probability and Statistics 5
= 119875( max1le119894le2(119898
119899+1)
1003816100381610038161003816120572119899(119910 + 119894119903
119899) minus 120572
119899(119910)
1003816100381610038161003816 ge 119872120598)
le
119862119901min119901
11199012+1199013
(119872120598)119901
(2120575)min119901
11199012+1199013le 120578
(43)
for any fixed 119910 isin [0 1] Since the upper bound does notdepend on 119910 and the probability measure as well as supre-mum function are continuous we obtain
119875( sup0le119910le1minus2120575
sup119910le119909le119910+2120575
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 ge 4119872120598 + 119862radic119899ℎ4
119899) le 120578
(44)
where 4119872120598 + 119862radic119899ℎ4
119899is arbitrarily small
Since condition (10) is checked the proof is completed
3 Tightness of the Standard Empirical Process
In this section we deal with the standard empirical processbuilt on an associated sequence of uniformly [0 1]distributedrvrsquos 119880
119895119895ge1
that is
120572119899(119909) =
1
radic119899
119899
sum
119895=1
(119868 [119880119895le 119909] minus 119909) (45)
where 119909 isin [0 1] We shall relax the restrictions imposedon the process by Yu in [7] to obtain tightness Preciselywe do not need stationarity any more due to the techniquedrawn from [14] and we lower the assumed rate at which thecovariance tends to zero
While proving tightness of the empirical process we willuse the criterion proved in the first section as well as some ofour covariance inequalities
We shall start with the fact known under the name ofmultinomial theorem We recall it in the following lemma
Lemma 2 For natural numbers119898 119899 and 1199091 119909
2 119909
119898isin R
(1199091+ sdot sdot sdot + 119909
119899)119898
= sum
1198961+sdotsdotsdot+119896
119899=119898
(119898
1198961 119896
119899
) prod
1le119894le119899
119909119896119894
119894 (46)
where (119898
1198961 119896
119899) = 119898(119896
1 sdot sdot sdot 119896
119899) and 119896
1 119896
119899isin
0 1 119898
In particular
(1199091+ sdot sdot sdot + 119909
119899)4
le 4 sdot sum
1198961+sdotsdotsdot+119896
119899=4
1199091198961
1sdot sdot sdot 119909
119896119899
119899 (47)
which implies for 119878119899= sum
119899
119894=1119883
119894 that
1198641198784
119899le 4 sdot sum
1198961+sdotsdotsdot+119896
119899=4
119864 (1198831198961
1sdot sdot sdot 119883
119896119899
119899)
le 4 sdot
119899
sum
119905=1
sum
0le119894119895119896le119899minus1
10038161003816100381610038161003816119864 (119883
119905119883
119905+119894119883
119905+119894+119895119883
119905+119894+119895+119896)10038161003816100381610038161003816
(48)
Let us now introduce the following notation due to Doukhanand Louhichi (see [14]) for a sequence of centered rvrsquos119883
119905119899
119899ge1
119862119903119902
= sup100381610038161003816100381610038161003816Cov (119883
1199051
sdot sdot sdot 119883119905119898
119883119905119898+119903sdot sdot sdot 119883
119905119902
)100381610038161003816100381610038161003816 (49)
where the supremum runs over all divisions of the groupcomposed of 119902 rvrsquos into two subgroups such that the distancebetween the highest index of the rvrsquos in the first group and thelowest index of the rvrsquos from the second group is equal to 119903119903 isin 1 2 For 119903 = 0 we shall define 119862
0119902= 1 Let us then
put
1198831199051
= 119868 [1198801199051
le 119909] minus 119909 1198831199052
= 119868 [1198801199052
le 119909] minus 119909 (50)
We will now estimate the summands in (48)If max119894 119895 119896 = 119894 then
10038161003816100381610038161003816119864 119883
119905sdot (119883
119905+119894119883
119905+119894+119895119883
119905+119894+119895+119896)10038161003816100381610038161003816
=100381610038161003816100381610038161003816Cov (119883
119905 119883
119905+119894119883
119905+119894+119895119883
119905+119894+119895+119896)100381610038161003816100381610038161003816le 119862
1198944
(51)
Similarly for max119894 119895 119896 = 119896 we have
10038161003816100381610038161003816119864 (119883
119905119883
119905+119894119883
119905+119894+119895) sdot 119883
119905+119894+119895+11989610038161003816100381610038161003816
=10038161003816100381610038161003816Cov (119883
119905119883
119905+119894119883
119905+119894+119895 119883
119905+119894+119895+119896)10038161003816100381610038161003816le 119862
1198964
(52)
When max119894 119895 119896 = 119895 then
10038161003816100381610038161003816119864 (119883
119905119883
119905+119894) sdot (119883
119905+119894+119895119883
119905+119894+119895+119896)10038161003816100381610038161003816
=10038161003816100381610038161003816Cov (119883
119905119883
119905+119894 119883
119905+119894+119895119883
119905+119894+119895+119896)
+119864 (119883119905119883
119905+119894) 119864 (119883
119905+119894+119895119883
119905+119894+119895+119896)10038161003816100381610038161003816
le 1198621198954
+ 1198621198942119862
1198962
(53)
We keep on estimating the fourth moment of 119878119899by intro-
ducing 119905 isin 0 119899mdashthe index for which |119864(119883119905119883
119905+119894119883
119905+119894+119895
119883119905+119894+119895+119896
)| attains its maximum
1198641198784
119899le 4119899 sum
0le119894119895119896le119899minus1
10038161003816100381610038161003816119864 (119883
119905119883
119905+119894119883
119905+119894+119895119883
119905+119894+119895+119896)10038161003816100381610038161003816
=4119899( sum
0le119894119896le119895
[1198621198942sdot 119862
1198962+ 119862
1198954]+ sum
0le119894119895le119896
1198621198964+ sum
0le119896119895le119894
1198621198944)
= 4119899( sum
0le119894119896le119895
1198621198942sdot 119862
1198962+ 3 sum
0le119896119895le119894
1198621198944)
(54)
6 ISRN Probability and Statistics
The terms obtained in (54) may further be bounded fromabove in the following way
sum
0le119894119896le119895
1198621198942sdot 119862
1198962=
119899minus1
sum
119895=0
119895
sum
119894119896=0
1198621198942119862
1198962
le 119899
119899minus1
sum
119894119896=0
1198621198942119862
1198962= 119899(
119899minus1
sum
119894=0
1198621198942)
2
sum
0le119896119895le119894
1198621198944
=
119899minus1
sum
119894=0
119894
sum
119895119896=0
1198621198944
le
119899minus1
sum
119894=0
(119894 + 1) (119894 + 1) 1198621198944
=
119899minus1
sum
119894=0
(119894 + 1)2119862
1198944
(55)
We thus get the inequality
1198641198784
119899le 4([119899
119899minus1
sum
119894=0
1198621198942]
2
+ 3119899
119899minus1
sum
119894=0
(119894 + 1)2119862
1198944) (56)
We shall now focus on estimating 1198621199032
and 1198621199034 Since 119880
1199051
and 1198801199052
are associated uniformly [0 1] distributed rvrsquos 1198831199051
and 1198831199052
mdashas monotone functions of these rvrsquosmdashare associat-ed as well In order to bound
1198621199032
= sup0le1199051lt1199052le119899 1199052minus1199051=119903
Cov (1198831199051
1198831199052
) (57)
let us notice that from Schwarz inequality
Cov (1198831199051
1198831199052
) = 119864 (119868 [1198801199051
le 119909] 119868 [1198801199052
le 119909]) minus 1199092
le radic1199092 minus 1199092le 119909
(58)
On the other hand invoking inequalities from [13 15]
Cov (1198831199051
1198831199052
)
le sup119909isin[01]
[119875 (1198801199051
le 1199091198801199052
le 119909) minus 119875 (1198801199051
le 119909)119875 (1198801199052
le 119909)]
le 119862 sdot Cov13
(1198801199051
1198801199052
) for associated rvrsquos4 sdot Cov (119880
1199051
1198801199052
) for MTP2rvrsquos
(59)
As a consequence
1198621199032
le min 119909 119862 sdot Cov13
(1198801199051
1198801199052
) for associated rvrsquosmin 119909 4 sdot Cov (119880
1199051
1198801199052
) for MTP2rvrsquos
(60)
where 1199052minus 119905
1= 119903 Still we need the upper bound for 119862
1199034 It
turns out that119862
1199034
= sup1003816100381610038161003816100381610038161003816100381610038161003816
Cov2
prod
119894=1
(119868 [119880119905119894
le 119909] minus 119909)
4
prod
119894=3
(119868 [119880119905119894
le 119909] minus 119909)
1003816100381610038161003816100381610038161003816100381610038161003816
(61)
In other words among all divisions of the group 119883119905119894
119894 isin
1 2 3 4 into two subgroups 1198621199034
attains the biggest valuein case we take two subgroups consisted of two rvrsquos Thesupremum runs over the set 119905
1 119905
2 119905
3 119905
4isin N 0 le 119905
1lt 119905
2lt
1199053lt 119905
4le 119899 and 119905
3minus 119905
2= 119903 Elementary calculation leads to the
following formula
1198621199034
= sup 10038161003816100381610038161003816119875 (119880119905119894
le 119909 119894 isin 1 2 3 4)
minus 119875 (119880119905119894
le 119909 119894 isin 1 2) 119875 (119880119905119894
le 119909 119894 isin 3 4)
minus 119909 [119875 (119880119905119894
le 119909 119894 isin 1 2 3)
minus 119875 (119880119905119894
le 119909 119894 isin 1 2) 119875 (1198801199053
le 119909)
+ 119875 (119880119905119894
le 119909 119894 isin 1 3 4)
minus 119875 (119880119905119894
le 119909 119894 isin 3 4) 119875 (1198801199051
le 119909)
+ 119875 (119880119905119894
le 119909 119894 isin 1 2 4)
minus 119875 (119880119905119894
le 119909 119894 isin 1 2) 119875 (1198801199054
le 119909)
+ 119875 (119880119905119894
le 119909 119894 isin 2 3 4)
minus119875 (119880119905119894
le 119909 119894 isin 3 4) 119875 (1198801199052
le 119909)]
+ 1199092[119875 (119880
1199051
le 1199091198801199053
le 119909)
minus 119875 (1198801199051
le 119909)119875 (1198801199053
le 119909)
+ 119875 (1198801199051
le 1199091198801199054
le 119909)
minus 119875 (1198801199051
le 119909)119875 (1198801199054
le 119909)
+ 119875 (1198801199052
le 1199091198801199053
le 119909)
minus 119875 (1198801199052
le 119909)119875 (1198801199053
le 119909)
+ 119875 (1198801199052
le 1199091198801199054
le 119909)
minus119875 (1198801199052
le 119909)119875 (1198801199052
le 119909)]10038161003816100381610038161003816
le10038161003816100381610038161198771
1003816100381610038161003816 +10038161003816100381610038161198772
1003816100381610038161003816 +10038161003816100381610038161198773
1003816100381610038161003816
(62)
where for the sake of simplicity 1198771 119877
2 and 119877
3are respec-
tively the free coefficient the expression with 119909 and theexpression with 119909
2 Using the Lebowitz inequality (see [15]for instance) and inequalities obtained in [13 15] we arrive at10038161003816100381610038161198771
1003816100381610038161003816 =10038161003816100381610038161003816119875 (119880
119905119894
le 119909 119894 isin 1 2 3 4)
minus119875 (119880119905119894
le 119909 119894 isin 1 2) 119875 (119880119905119894
le 119909 119894 isin 3 4)10038161003816100381610038161003816
le 11986711988011990511198801199053
+ 11986711988011990511198801199054
+ 11986711988011990521198801199053
+ 11986711988011990521198801199054
ISRN Probability and Statistics 7
le
119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
)) for associated rvrsquos
4(
2
sum
119894=1
4
sum
119895=3
Cov (119880119905119894
119880119905119895
)) for MTP2rvrsquos
(63)
Analogously we get
10038161003816100381610038161198772
1003816100381610038161003816
le
2119909 sdot 119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
)) for associated rvrsquos
2119909 sdot 4(
2
sum
119894=1
4
sum
119895=3
Cov (119880119905119894
119880119905119895
)) for MTP2rvrsquos
10038161003816100381610038161198773
1003816100381610038161003816
le
1199092sdot 119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
) ) for associated rvrsquos
1199092sdot 4(
2
sum
119894=1
4
sum
119895=3
Cov (119880119905119894
119880119905119895
)) for MTP2rvrsquos
(64)
Eventually we have
1198621199034
le
119891 (119909) sdot119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
)) for associated rvrsquos
119891 (119909) sdot4(
2
sum
119894=1
4
sum
119895=3
Cov (119880119905119894
119880119905119895
)) for MTP2rvrsquos
(65)
where 1199053minus 119905
2= 119903 and 119891(119909) = (1 + 2119909 + 119909
2) for 119909 isin [0 1]
Let us now introduce the following notation
120579119903=
sup119905isinN
Cov (119880119905 119880
119905+119903) for 119903 isin 1 2
1 for 119903 = 0
(66)
and assume it decays powerly at rate 119886 in the following way
120579119903= 119874(
1
(119903 + 1)119886) for 119886 gt 0 119903 isin 0 1 (67)
Let us get back to inequality (56) we can now carry on Atfirst for associated rvrsquos
1198641198784
119899le 4[119899
119899minus1
sum
119903=0
min 119909 119862 sdot Cov13(119880
1199051
1198801199052
)]
2
+ 4 sdot 3119899
119899minus1
sum
119903=0
(119903 + 1)2119891 (119909) sdot 119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
))
le 119863 sdot ([119899
119899minus1
sum
119903=0
min 119909 (119903 + 1)minus1198863
]
2
+119899
119899minus1
sum
119903=0
(119903 + 1)2(119903 + 1)
minus1198863)
= 119863 sdot ([119899
119899
sum
119903=1
min 119909 119903minus1198863]
2
+ 119899
119899
sum
119903=1
1199032119903minus1198863
)
le 119863 sdot ([
[
119899 sum
119903lt119909minus3119886
119909 + 119899 sum
119903ge119909minus3119886
1
1199031198863]
]
2
+ 119899
119899
sum
119903=1
1199032minus1198863
)
le 1198631sdot (119899
2119909
2(119886minus3)119886+ 120585)
(68)
where119863 and1198631are constants and
120585 = 119899
119899
sum
119903=1
1199032minus1198863
=
119874(119899) 119886 gt 9
119874 (119899 ln 119899) 119886 = 9
119874 (1198994minus1198863
) 3 lt 119886 lt 9
(69)
It is worth mentioning that in the last inequality of (68) weused the estimate
int
infin
119909
1
119905119901119889119905 sim
1
119909119901minus1for 119901 gt 1 (70)
At the same time in the case of MTP2rvrsquos we get
1198641198784
119899le 119863
2sdot (119899
2119909
2(119886minus1)119886+ 120577) (71)
where1198632is constant and
120577 = 119899
119899
sum
119903=1
1199032minus119886
=
119874(119899) 119886 gt 3
119874 (119899 ln 119899) 119886 = 3
119874 (1198994minus119886
) 1 lt 119886 lt 3
(72)
Let now 120572119899(119909) = (1radic119899)sum
119899
119894=1(119868[119880
119905119894
le 119909] minus 119909) Then
120572119899(119909) minus 120572
119899(119910) =
1
radic119899
119899
sum
119894=1
(119868 [119909 lt 119880119905119894
le 119910] minus (119909 minus 119910))
for 119909 119910 isin [0 1]
(73)
8 ISRN Probability and Statistics
has the fourth moment estimatedmdashin the case of associatedrvrsquosmdashby
119864[120572119899(119909) minus 120572
119899(119910)]
4
= 119864[1
radic119899
119899
sum
119894=1
(119868 [119909 lt 119880119905119894
le 119910] minus (119909 minus 119910))]
4
=1
1198992119864[
119899
sum
119894=1
(119868 [119909 lt 119880119905119894
le 119910] minus (119909 minus 119910))]
4
le 1198631sdot (
1
119899211989921003816100381610038161003816119909 minus 119910
1003816100381610038161003816
2(119886minus3)119886
+120585
1198992)
= 1198631sdot (
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2(119886minus3)119886
+120585
1198992)
=
119874(119899minus1) 119886 gt 9
119874(ln 119899119899
) 119886 = 9
119874 (1198992minus1198863
) 3 lt 119886 lt 9
(74)
and in the case of MTP2rvrsquos by
119864[120572119899(119909) minus 120572
119899(119910)]
4
le 1198632sdot (
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2(119886minus1)119886
+120577
1198992)
=
119874(119899minus1) 119886 gt 3
119874(ln 119899119899
) 119886 = 3
119874 (1198992minus119886
) 1 lt 119886 lt 3
(75)
In light of the Shao and Yursquos criterion our process is tight forassociated rvrsquos when 119886 gt 6 and for MTP
2rvrsquos when 119886 gt 2
Let us sum up this result in the following theorem
Theorem 3 Let 120572119899(119909) = (1radic119899)sum
119899
119894=1(119868[119880
119894le 119909] minus 119909) be the
empirical process built on an associated sequence of uniformly[0 1] distributed rvrsquos 119880
119894119894ge1
Let also
120579119903= sup
119905isinNCov (119880
119905 119880
119905+119903) = 119874 ((119903 + 1)
minus119886)
where 119903 isin 0 1 119886 gt 0
(76)
Then 120572119899(119909)
119899ge1is tight for 119886 gt 6 If the rvrsquos 119880
119894119894ge1
are MTP2
then the process is tight for 119886 gt 2
Yu assumed stationarity of 119880119894119894ge1
andsum
infin
119899=111989965+]Cov(119880
0 119880
119899) lt infin for a positive constant ]
thus the rate of decay 119886 gt 7 5 Our result weakens consid-erably these assumptions especially in the case of MTP
2rvrsquos
Louhichi in [16] proposed a different tightness criterioninvolving the so-called bracketing numbers She managedto enhance Yursquos resultmdasheven more than Shao and Yu in[8]mdashsince she proved that it suffices to take 119886 gt 4 to gettightness of the empirical process based on the associated
rvrsquos Nevertheless she kept the assumption of stationarityvalid
In the final analysis our resultrsquos advantage is the absenceof the stationarity assumption and the rate of decay for120579119903remains (up to the authorrsquos knowledge) unimproved for
MTP2rvrsquos
Unfortunately with a view to obtainingweak convergenceof the process in question that is also convergence offinite-dimensional distributions we do not know how tomanage without the assumption of stationarityTherefore weconclude with the following corollary
Corollary 4 Let 120572119899(119909) = (1radic119899)sum
119899
119894=1(119868[119880
119894le 119909] minus 119909) be the
empirical process built on a stationary associated sequence ofuniformly [0 1] distributed rvrsquos 119880
119894119894ge1
Let also
120579119903= Cov (119880
1 119880
1+119903) = 119874 ((119903 + 1)
minus119886)
where 119903 isin 0 1 119886 gt 0
(77)
Then if 119886 gt 6
120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (78)
where 119861(sdot) is the zero mean Gaussian process on [0 1] withcovariance structure defined by
1205902(119909 119910) = 119909 and 119910 minus 119909119910
+
infin
sum
119895=1
[ Cov (119868 [1198801le 119909] 119868 [119880
119895+1le 119910])
+ Cov (119868 [1198801le 119910] 119868 [119880
119895+1le 119909])]
(79)
If the rvrsquos 119880119894119894ge1
are MTP2 then it suffices to be 119886 gt 2 in
order to claim the above convergence
Proof It remains to establish convergence of finite-dimen-sional distributions repeating the procedure from [7]
4 Tightness of the Kernel-TypeEmpirical Process
In this section we shall weaken assumption imposed on thecovariance structure of rvrsquos 119883
119895119895ge1
by Cai and Roussas in [1]for the kernel estimator of the df
119865119899(119909) =
1
119899
119899
sum
119895=1
119870(
119909 minus 119883119895
ℎ119899
) (80)
They deal with a stationary sequence of negatively associatedrvrsquos (cf [17]) and need the same condition as Yu [7] that is
1003816100381610038161003816Cov (1198831 119883
1+119899)1003816100381610038161003816 = 119874(
1
11989975+]) (81)
to get tightness of the smooth empirical process (see condi-tion (A4) in [1])
ISRN Probability and Statistics 9
It turns out that it suffices to have
1003816100381610038161003816Cov (1198801 119880
1+119899)1003816100381610038161003816 = 119874(
1
1198993119901(119901minus1)) (82)
where 119901 gt 4 is a positive constant taken from the tightnesscriterion (8) It is easy to see that asymptotically we get therate 3
On the way to prove it we will also take use of aRosenthal-type inequality due to Shao andYu (seeTheorem 2in [8]) we shall recall in the following lemma
Lemma 5 Let 119901 gt 2 and 119891 be a real valued function boundedby 1 with bounded first derivative Suppose that 119883
119899119899ge1
is asequence of stationary and associated rvrsquos such that for 119899 isin N
Cov (1198831 119883
119899) = 119874 (119899
minus119887) for some 119887 gt 119901 minus 1 (83)
Then for any 120583 gt 0 there exists some positive constant 119896120583
independent of the function 119891 for which
1198641003816100381610038161003816119878119899
(119891) minus 119864119878119899(119891)
1003816100381610038161003816
119901
le 119896120583(119899
1+12058310038171003817100381710038171003817119891
101584010038171003817100381710038171003817
2
+1198991199012 [
[
119899
sum
119895=1
10038161003816100381610038161003816Cov (119891 (119883
1)119891 (119883
119895))10038161003816100381610038161003816]
]
)
(84)
As we can see the lemma assumes association but itworks for negatively associated rvrsquos as well since in the proofit reaches back the result of Newman (see Proposition 15 in[18]) where both types of association are allowed
Let us recall that 120572119899(119909) = radic119899(119865
119899(119909) minus 119864119865
119899(119909)) where
119865119899(119909) = (1119899)sum
119899
119895=1119870((119909 minus 119883
119895)ℎ
119899) It is easy to see that
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
= 119899minus1199012
11986410038161003816100381610038161003816119878119899(119891 (119883
119895)) minus 119864119878
119899(119891 (119883
119895))10038161003816100381610038161003816
119901
(85)
where
119891 (119883119895) = 119870(
119909 minus 119883119895
ℎ119899
) minus 119870(
119910 minus 119883119895
ℎ119899
)
119878119899(119891) =
119899
sum
119895=1
119891 (119883119895)
(86)
With an intent to use Lemma 5 we need 119891 to be boundedfrom above by 1 (which in our case is obvious) and to havebounded 119891
1015840 thus we assume
sup119905isinR
10038161003816100381610038161003816119870
1015840(119905)
10038161003816100381610038161003816
1
ℎ119899
le119862
119870
ℎ119899
lt infin (87)
Now applying inequality (84) we have
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
le119896120583119899minus1199012
1198991+120583
1198622
119870
ℎ2
119899
+1198991199012[
[
119899
sum
119895=1
10038161003816100381610038161003816Cov (119891 (119883
1)119891 (119883
119895))10038161003816100381610038161003816]
]
1199012
(88)
Newman showed in [18] that
Cov (1198911(119883) 119891
2(119884))
= ∬R2119891
1015840
1(119909) 119891
1015840
2(119910)
times [119875 (119883 le 119909 119884 le 119910) minus 119875 (119883 le 119909) 119875 (119884 le 119910)] 119889119909 119889119910
(89)
if 1198911and 119891
2are real valued functions on R having square
integrable derivatives 1198911015840
1and 119891
1015840
2 respectively and provided
that 1198911(119883) 119891
2(119884) have finite secondmoments In light of that
equation and linearity of covariance we get10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
=
1003816100381610038161003816100381610038161003816∬
R2[119875 (119883
1lt 119909 minus ℎ
119899119904 119883
119895lt 119909 minus ℎ
119899119905)
minus119875 (1198831lt 119909 minus ℎ
119899119904 119883
119895lt 119910 minus ℎ
119899119905)]
minus [119875 (1198831lt 119909 minus ℎ
119899119904) 119875 (119883
119895lt 119909 minus ℎ
119899119905)
minus119875 (1198831lt 119909 minus ℎ
119899119904) 119875 (119883
119895lt 119910 minus ℎ
119899119905)]
minus [119875 (1198831lt 119910 minus ℎ
119899119904 119883
119895lt 119909 minus ℎ
119899119905)
minus119875 (1198831lt 119910 minus ℎ
119899119904 119883
119895lt 119910 minus ℎ
119899119905)]
minus [119875 (1198831lt 119910 minus ℎ
119899119904) 119875 (119883
119895lt 119910 minus ℎ
119899119905)
minus 119875 (1198831lt 119910 minus ℎ
119899119904)
times 119875 (119883119895lt 119909 minus ℎ
119899119905)] 119896 (119904) 119896 (119905) 119889119904 119889119905
10038161003816100381610038161003816100381610038161003816
(90)
Without the loss of generality we may and do assume that119909 gt 119910 thus10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
=
1003816100381610038161003816100381610038161003816∬
R2119875 (119910 minus ℎ
119899119904 le 119883
1lt 119909 minus ℎ
119899119904
119910 minus ℎ119899119905 le 119883
119895lt 119909 minus ℎ
119899119905)
minus 119875 (119910 minus ℎ119899119904 le 119883
1lt 119909 minus ℎ
119899119904)
times 119875 (119910 minus ℎ119899119905 le 119883
119895lt 119909 minus ℎ
119899119905) 119896 (119904) 119896 (119905) 119889119904 119889119905
1003816100381610038161003816100381610038161003816
(91)
and by triangle inequality we get10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
le ∬R2[int
119909minusℎ119899119904
119910minusℎ119899119904
int
119909minusℎ119899119905
119910minusℎ119899119905
1198911198831119883119895(119906 V) 119889119906 119889V
+int
119909minusℎ119899119904
119910minusℎ119899119904
1198911198831(119906) 119889119906int
119909minusℎ119899119905
119910minusℎ119899119905
119891119883119895(V) 119889V] 119896 (119904) 119896 (119905) 119889119904 119889119905
(92)
10 ISRN Probability and Statistics
1198911198831119883119895
(119906 V) 1198911198831
(119906) and 119891119883119895
(V) are the joint pdf of [1198831 119883
119895]
and marginal pdfrsquos of rvrsquos1198831and119883
119895 respectively We need
to further assume that 1198621stands for a common upper bound
of 1198911198831
and 1198911198831119883119895
for all 119895 isin N that is
100381710038171003817100381710038171198911198831
10038171003817100381710038171003817le 119862
1
1003817100381710038171003817100381710038171198911198831119883119895
100381710038171003817100381710038171003817le 119862
1forall119895 ge 1 (93)
Then we finally obtain
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816le 2119862
1
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
where = max 1198621 119862
2
1
(94)
On the other hand proceeding like Cai and Roussas in [1] wecan shortly get
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
le
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119909 minus 119883
1
ℎ119899
) 119870(
119909 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119909 minus 119883
1
ℎ119899
) 119870(
119910 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119910 minus 119883
1
ℎ119899
) 119870(
119909 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119910 minus 119883
1
ℎ119899
) 119870(
119910 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
le 41198622
10038161003816100381610038161003816Cov (119883
1 119883
119895)10038161003816100381610038161003816
13
(95)
where 1198622is a constant relevant to the covariance inequality
for negatively associated rvrsquos (see [13])We now arrive at the following inequality
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
le max 1198621 119862
2
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
10038161003816100381610038161003816Cov(119883
1 119883
119895)10038161003816100381610038161003816
13
(96)
Assuming |Cov(119891(1198831) 119891(119883
119895))| = 119874(119895
minus119903) and proceeding
similarly to (68) we can get
[
[
119899
sum
119895=1
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816]
]
1199012
le max 1198621 119862
2 [
[
sum
119895lt|119909minus119910|minus2(1199033)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
+ sum
119895ge|119909minus119910|minus2(1199033)
1
1198951199033]
]
1199012
le 2max 1198621 119862
21003816100381610038161003816119909 minus 119910
1003816100381610038161003816
((119903minus3)119903)119901
(97)
To sum up we obtain the following inequality
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
le 119896120583119899
1+120583minus1199012119862
2
119870
ℎ2
119899
+ 2max 1198621 119862
21003816100381610038161003816119909 minus 119910
1003816100381610038161003816
((119903minus3)119903)119901
le 1198621198991+120583minus1199012 1
ℎ2
119899
+1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
((119903minus3)119903)119901
(98)
where 119862 = 119896120583max1198622
119896 2max119862
1 119862
2 The formula of the
tightness criterion (8) implies that ((119903 minus 3)119903)119901 gt 1 so
119903 gt3119901
119901 minus 1 where 119901 gt 2 (99)
Let us recall the assumptions imposed on the bandwidhtsℎ
119899119899ge1
by Cai and Roussas in [1]B1 lim
119899rarrinfinℎ119899= 0 and ℎ
119899gt 0 for all 119899 isin N
+
B2 lim119899rarrinfin
119899ℎ119899= infin thus ℎ
119899= 119874(1119899
1minus120573) 120573 gt 0
B3 lim119899rarrinfin
119899ℎ4
119899= 0 hence ℎ
119899= 119874(1119899
14+120575) 120575 gt 0
In light of the above
1198991+120583minus1199012 1
ℎ2
119899
= 11989932+120583+2120575minus1199012
(100)
where 120583 gt 0 119901 gt 2 and 0 lt 120575 lt 34 Confrontation with thetightness criterion (8) forces
3
2+ 120583 + 2120575 minus
119901
2lt minus
1
2 (101)
which implies 119901 gt 4 Let us conclude with the followingtheorem
Theorem6 Let 120572119899(119909) = radic119899(119865
119899(119909)minus119864119865
119899(119909)) be the empirical
process built on the kernel estimator of the df119865 for a stationarysequence of negatively associated rvrsquos Assume that conditionsA1 A2 (87) B1 B2 B3 are satisfied Then provided that thetightness criterion (8) holds with 119901 gt 4 it suffices to demand
1003816100381610038161003816Cov (1198801 119880
1+119899)1003816100381610038161003816 = 119874(
1
1198993119901(119901minus1)) (102)
in order to obtain120572
119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (103)
where 119861(sdot) is the zero mean Gaussian process with covariancestructure defined by
1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)
+
infin
sum
119895=1
[ Cov (119868 [1198831le 119909] 119868 [119883
119895+1le 119910])
+ Cov (119868 [1198831le 119910] 119868 [119883
119895+1le 119909])]
(104)
and 119909 119910 isin [0 1]
Proof The remaining covergence of finite-dimensional dis-tributions of 120572
119899(119909) is established in [1]
ISRN Probability and Statistics 11
5 Weak Convergence of theRecursive Kernel-Type Empirical Processunder IID Assumption
The aim of this section is to show weak convergence of theempirical process
120572119899(119909) =
1
radic119899
119899
sum
119895=1
[119870(
119909 minus 119883119895
ℎ119895
) minus 119864119870(
119909 minus 119883119895
ℎ119895
)] (105)
built on iid rvrsquos 119883119895119895ge1
Wewill prove that120572
119899(sdot) convergesweakly to the Brownian
bridge 119861(sdot) with the following covariance structure
1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910) for 119909 119910 isin [0 1]
(106)
It turns out that the tightness criterion (8) suffices to reachthe goal Convergence of finite-dimensional distributions of120572
119899(sdot) holds and we shall show itProceeding like Cai and Roussas in [1] we need to show
that for any 119886 119887 isin R
119886120572119899(119909) + 119887120572
119899(119910) 997888rarr 119886119861 (119909) + 119887119861 (119910) in distribution
(107)
Let us introduce the notation
119884119894(119909) = 119870(
119909 minus 119883119894
ℎ119894
) minus 119864119870(119909 minus 119883
119894
ℎ119894
) (108)
and look into the covariance structure of 120572119899(sdot)
Cov (120572119899(119909) 120572
119899(119910))
= Cov( 1
radic119899
119899
sum
119894=1
119884119894(119909)
1
radic119899
119899
sum
119894=1
119884119894(119910))
=1
119899
119899
sum
119894=1
Cov (119884119894(119909) 119884
119894(119910))
=1
119899
119899
sum
119894=1
Cov(119870(119909 minus 119883
119894
ℎ119894
) 119870(119910 minus 119883
119894
ℎ119894
))
(109)
where the second equality follows from assumed indepen-dence of rvrsquos 119883
119894119894ge1
Firstly we observe that each summand converges to119865(119909and
119910) minus 119865(119909)119865(119910) since
Cov(119870(119909 minus 119883
119894
ℎ119894
) 119870(119910 minus 119883
119894
ℎ119894
))
= 119864[119870(119909 minus 119883
119894
ℎ119894
)119870(119910 minus 119883
119894
ℎ119894
)]
minus 119864119870(119909 minus 119883
119894
ℎ119894
)119864119870(119910 minus 119883
119894
ℎ119894
)
(110)
where
119864[119870(119909 minus 119883
119894
ℎ119894
)119870(119910 minus 119883
119894
ℎ119894
)]
= int
119909and119910
minusinfin
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
+ int
119909or119910
119909and119910
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
+ int
infin
119909or119910
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
(111)
and 119865 denotes the common df of the rvrsquos 119883119894119894ge1
In [6] itwas shown that
119864119870(119909 minus 119883
119894
ℎ119894
) 997888rarr 119865 (119909) (112)
and recalling that the kernel function 119870 is a df as well wearrive at the conclusion
Secondly applying Toeplitz lemma we get
Cov (120572119899(119909) 120572
119899(119910)) 997888rarr 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)
= 1205902(119909 119910) 119899 997888rarr infin
(113)
Thus
Var (119886120572119899(119909) + 119887120572
119899(119910))
= 1198862 Var (120572
119899(119909)) + 2119886119887Cov (120572
119899(119909) 120572
119899(119910))
+ 1198872 Var (120572
119899(119910))
997888rarr 1198862120590
2(119909 119909) + 2119886119887120590
2(119909 119910) + 119887
2120590
2(119910 119910) 119899 997888rarr infin
(114)
We are now on the way to prove that 119886120572119899(119909) + 119887120572
119899(119910) con-
verges in distribution to 119886119861(119909) + 119887119861(119910) sim N(0 1198862120590
2(119909 119909) +
21198861198871205902(119909 119910) + 119887
2120590
2(119910 119910)) which in other words means that
1
radic119899
infin
sum
119894=1
119886119884119894(119909) + 119887119884
119894(119910)
radicVar (119886120572119899(119909) + 119887120572
119899(119910))
997888rarr N (0 1) (115)
In order to obtain (115) we will use Lyapunov condition
lim119899rarrinfin
1
1198632+120575
119899
119899
sum
119894=1
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884119894(119910)
1003816100381610038161003816
2+120575
= 0 (116)
for 120575 = 1 where 1198632
119899= Var(sum119899
119894=1[119886119884
119894(119909) + 119887119884
119894(119910)]) Since
|119870(sdot) minus 119864119870(sdot)| le 2
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884i (119910)1003816100381610038161003816
3
le 8 (1198863+ 119887
3) (117)
Moreover 1198632
119899= 119899Var(119886120572
119899(119909) + 119887120572
119899(119910)) together with (114)
yields
1198633
119899= 119874 (119899
32) (118)
12 ISRN Probability and Statistics
Combining (117) with (118) we obtain
1
1198633
119899
119899
sum
119894=1
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884119894(119910)
1003816100381610038161003816
3
= 119874 (119899minus12
) (119)
which shows that Lyapunov condition holds and completesthe proof of convergence of finite-dimensional distributionsof 120572
119899(119909)We summarize the result of that section in the following
theorem
Theorem 7 Let 120572119899(119909) be the recursive kernel-type empirical
process defined by (105) built on iid rvrsquos 119883119895119895ge1
withmarginal df 119865 If 120572
119899(sdot) satisfies the tightness criterion (8) then
120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (120)
where 119861 is the Brownian bridge
Acknowledgment
The author would like to thank the referees for the carefulreading of the paper and helpful remarks
References
[1] Z Cai and G G Roussas ldquoWeak convergence for smoothestimator of a distribution function under negative associationrdquoStochastic Analysis and Applications vol 17 no 2 pp 145ndash1681999
[2] Z Cai andG G Roussas ldquoEfficient Estimation of a DistributionFunction under Qudrant Dependencerdquo Scandinavian Journal ofStatistics vol 25 no 1 pp 211ndash224 1998
[3] ZW Cai and G G Roussas ldquoUniform strong estimation under120572-mixing with ratesrdquo Statistics amp Probability Letters vol 15 no1 pp 47ndash55 1992
[4] G G Roussas ldquoKernel estimates under association stronguniform consistencyrdquo Statistics amp Probability Letters vol 12 no5 pp 393ndash403 1991
[5] Y F Li and S C Yang ldquoUniformly asymptotic normalityof the smooth estimation of the distribution function underassociated samplesrdquo Acta Mathematicae Applicatae Sinica vol28 no 4 pp 639ndash651 2005
[6] Y Li C Wei and S Yang ldquoThe recursive kernel distributionfunction estimator based onnegatively and positively associatedsequencesrdquo Communications in Statistics vol 39 no 20 pp3585ndash3595 2010
[7] H Yu ldquoA Glivenko-Cantelli lemma and weak convergence forempirical processes of associated sequencesrdquo ProbabilityTheoryand Related Fields vol 95 no 3 pp 357ndash370 1993
[8] Q M Shao and H Yu ldquoWeak convergence for weightedempirical processes of dependent sequencesrdquo The Annals ofProbability vol 24 no 4 pp 2098ndash2127 1996
[9] P Billingsley Convergence of Probability Measures Wiley Seriesin Probability and Statistics Probability and Statistics JohnWiley amp Sons New York NY USA 1999
[10] J D Esary F Proschan and D W Walkup ldquoAssociation ofrandom variables with applicationsrdquo Annals of MathematicalStatistics vol 38 pp 1466ndash1474 1967
[11] A Bulinski and A Shashkin Limit Theorems for AssociatedRandom Fields and Related Systems vol 10 of Advanced Serieson Statistical Science and Applied Probability World Scientific2007
[12] S Karlin and Y Rinott ldquoClasses of orderings of measures andrelated correlation inequalities I Multivariate totally positivedistributionsrdquo Journal of Multivariate Analysis vol 10 no 4 pp467ndash498 1980
[13] P Matuła and M Ziemba ldquoGeneralized covariance inequali-tiesrdquo Central European Journal of Mathematics vol 9 no 2 pp281ndash293 2011
[14] P Doukhan and S Louhichi ldquoA new weak dependence condi-tion and applications to moment inequalitiesrdquo Stochastic Proc-esses and their Applications vol 84 no 2 pp 313ndash342 1999
[15] P Matuła ldquoA note on some inequalities for certain classes ofpositively dependent random variablesrdquo Probability and Math-ematical Statistics vol 24 no 1 pp 17ndash26 2004
[16] S Louhichi ldquoWeak convergence for empirical processes of asso-ciated sequencesrdquo Annales de lrsquoInstitut Henri Poincare vol 36no 5 pp 547ndash567 2000
[17] K Joag-Dev and F Proschan ldquoNegative association of randomvariables with applicationsrdquoThe Annals of Statistics vol 11 no1 pp 286ndash295 1983
[18] C M Newman ldquoAsymptotic independence and limit theoremsfor positively and negatively dependent random variablesrdquo inInequalities in Statistics and Probability Y L Tong Ed vol 5pp 127ndash140 Institute ofMathematical StatisticsHayward CalifUSA 1984
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 ISRN Probability and Statistics
Since 119870 is a nondecreasing function and applying triangleinequality again we get
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816
le1003816100381610038161003816120572119899
(119910 + 119902) minus 120572119899(119910)
1003816100381610038161003816 +radic119899
10038161003816100381610038161003816119864119865
119899(119910 + 119902) minus 119864119865
119899(119910)
10038161003816100381610038161003816
+ radic119899 (10038161003816100381610038161003816119864119865
119899(119909) minus 119865 (119909)
10038161003816100381610038161003816+10038161003816100381610038161003816119864119865
119899(119910) minus 119865 (119910)
10038161003816100381610038161003816
+1003816100381610038161003816119865 (119909) minus 119865 (119910)
1003816100381610038161003816 )
(33)
Cai and Roussas in [1] showed that under assumptions madeon the df 119865 and the kernel function119870 we have
10038161003816100381610038161003816119864119865
119899(119909) minus 119865 (119909)
10038161003816100381610038161003816= 119874 (ℎ
2
119899) (34)
Similarly by Taylor expansion it is easy to see that
1003816100381610038161003816119865 (119909) minus 119865 (119910)1003816100381610038161003816 le
10038171003817100381710038171198911003817100381710038171003817 sdot
1003816100381610038161003816119909 minus 1199101003816100381610038161003816 (35)
Thus
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 le1003816100381610038161003816120572119899
(119910 + 119902) minus 120572119899(119910)
1003816100381610038161003816
+ 119862radic119899ℎ4
119899+119872radic119899119902
(36)
where 119872 = sup119909isin[01]
119891(119909) and 119862 = 1198621198911015840119870
is a positive con-stant dependent on the functions 119865 and119870
Let us recall that 119910 le 119909 le 119910+ 119902119898119899= lfloor120575119903
119899rfloor and 119902 ge 120598119899
Taking 119902 = (120598radic119899)(= 119903119899) we notice that 119898
119899119902 = lfloor120575119903
119899rfloor119903
119899le 120575
thus when 119899 rarr infin the interval [119910 119910 + 119898119899119902] coincides with
[119910 119910 + 120575] Approaching the aim of Step 4 let us observe that
sup119910le119909le119910+119898
119899119902
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 = max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572
119899(119910)
1003816100381610038161003816
+1003816100381610038161003816120572119899
(1199090) minus 120572
119899(119910 + 119894max119902)
1003816100381610038161003816
(37)
where 1199090isin [119910+119894
0119902 119910+(119894
0+1)119902] is a point at which the above
supremum is attained and
119894max =
1198940+ 1 if 1003816100381610038161003816120572119899
(119910 + [1198940+ 1] 119902) minus 120572
119899(119910)
1003816100381610038161003816
ge1003816100381610038161003816120572119899
(119910 + 1198940119902) minus 120572
119899(119910)
1003816100381610038161003816
1198940
otherwise(38)
Without the loss of generality we may assume that 119894max = 1198940+
1 which implies
sup119910le119909le119910+119898
119899119902
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 = max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572
119899(119910)
1003816100381610038161003816
+1003816100381610038161003816120572119899
(1199090) minus 120572
119899(119910 + 119894
0119902)1003816100381610038161003816
(39)
Now applying inequality (36) the definition of 119894max and thetriangle inequality we have
sup119910le119909le119910+119898
119899119902
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816
le max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572
119899(119910)
1003816100381610038161003816
+1003816100381610038161003816120572119899
(119910 + [1198940+ 1] 119902) minus 120572
119899(119910 + 119894
0119902)1003816100381610038161003816
+ 119872radic119899119902 + 119862radic119899ℎ4
119899
le max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572
119899(119910)
1003816100381610038161003816
+1003816100381610038161003816120572119899
(119910 + [1198940+ 1] 119902) minus 120572
119899(119910)
1003816100381610038161003816
+1003816100381610038161003816120572119899
(119910 + 1198940119902) minus 120572
119899(119910)
1003816100381610038161003816 + 119872radic119899119902 + 119862radic119899ℎ4
119899
le 3 max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572
119899(119910)
1003816100381610038161003816 + 119872radic119899119902 + 119862radic119899ℎ4
119899
(40)
If we plug in 119902 = 119903119899we arrive at
sup119910le119909le119910+119898
119899119903119899
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 le 3 max1le119894le119898
119899
1003816100381610038161003816120572119899(119910 + 119894119903
119899) minus 120572
119899(119910)
1003816100381610038161003816
+ 119872radic119899119903119899+ 119862radic119899ℎ4
119899
(41)
Step 5 Finally we are in a position to obtain inequality (23)that is
forall120598gt0forall
120578gt0exist
0le120575lt12exist
1198990
forall119899ge1198990
119875( sup0le119910le1minus2120575
sup119910le119909le119910+2120575
1003816100381610038161003816120572119899(119909)minus120572
119899(119910)
1003816100381610038161003816
ge 120598) le 120578
(42)
We now successivelymake use the inequalities (29) (41) (27)(29) and (28) to get
119875( sup119910le119909le119910+2120575
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 ge 4119872120598 + 119862radic119899ℎ4
119899)
le 119875( sup119910le119909le119910+2(119898
119899+1)119903119899
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 ge 4119872120598 + 119862radic119899ℎ4
119899)
le 119875(3 max1le119894le2(119898119899+1)
1003816100381610038161003816120572119899(119910 + 119894119903
119899) minus 120572
119899(119910)
1003816100381610038161003816
+119872radic119899119903119899+ 119862radic119899ℎ4
119899ge 4119872120598 + 119862radic119899ℎ4
119899)
ISRN Probability and Statistics 5
= 119875( max1le119894le2(119898
119899+1)
1003816100381610038161003816120572119899(119910 + 119894119903
119899) minus 120572
119899(119910)
1003816100381610038161003816 ge 119872120598)
le
119862119901min119901
11199012+1199013
(119872120598)119901
(2120575)min119901
11199012+1199013le 120578
(43)
for any fixed 119910 isin [0 1] Since the upper bound does notdepend on 119910 and the probability measure as well as supre-mum function are continuous we obtain
119875( sup0le119910le1minus2120575
sup119910le119909le119910+2120575
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 ge 4119872120598 + 119862radic119899ℎ4
119899) le 120578
(44)
where 4119872120598 + 119862radic119899ℎ4
119899is arbitrarily small
Since condition (10) is checked the proof is completed
3 Tightness of the Standard Empirical Process
In this section we deal with the standard empirical processbuilt on an associated sequence of uniformly [0 1]distributedrvrsquos 119880
119895119895ge1
that is
120572119899(119909) =
1
radic119899
119899
sum
119895=1
(119868 [119880119895le 119909] minus 119909) (45)
where 119909 isin [0 1] We shall relax the restrictions imposedon the process by Yu in [7] to obtain tightness Preciselywe do not need stationarity any more due to the techniquedrawn from [14] and we lower the assumed rate at which thecovariance tends to zero
While proving tightness of the empirical process we willuse the criterion proved in the first section as well as some ofour covariance inequalities
We shall start with the fact known under the name ofmultinomial theorem We recall it in the following lemma
Lemma 2 For natural numbers119898 119899 and 1199091 119909
2 119909
119898isin R
(1199091+ sdot sdot sdot + 119909
119899)119898
= sum
1198961+sdotsdotsdot+119896
119899=119898
(119898
1198961 119896
119899
) prod
1le119894le119899
119909119896119894
119894 (46)
where (119898
1198961 119896
119899) = 119898(119896
1 sdot sdot sdot 119896
119899) and 119896
1 119896
119899isin
0 1 119898
In particular
(1199091+ sdot sdot sdot + 119909
119899)4
le 4 sdot sum
1198961+sdotsdotsdot+119896
119899=4
1199091198961
1sdot sdot sdot 119909
119896119899
119899 (47)
which implies for 119878119899= sum
119899
119894=1119883
119894 that
1198641198784
119899le 4 sdot sum
1198961+sdotsdotsdot+119896
119899=4
119864 (1198831198961
1sdot sdot sdot 119883
119896119899
119899)
le 4 sdot
119899
sum
119905=1
sum
0le119894119895119896le119899minus1
10038161003816100381610038161003816119864 (119883
119905119883
119905+119894119883
119905+119894+119895119883
119905+119894+119895+119896)10038161003816100381610038161003816
(48)
Let us now introduce the following notation due to Doukhanand Louhichi (see [14]) for a sequence of centered rvrsquos119883
119905119899
119899ge1
119862119903119902
= sup100381610038161003816100381610038161003816Cov (119883
1199051
sdot sdot sdot 119883119905119898
119883119905119898+119903sdot sdot sdot 119883
119905119902
)100381610038161003816100381610038161003816 (49)
where the supremum runs over all divisions of the groupcomposed of 119902 rvrsquos into two subgroups such that the distancebetween the highest index of the rvrsquos in the first group and thelowest index of the rvrsquos from the second group is equal to 119903119903 isin 1 2 For 119903 = 0 we shall define 119862
0119902= 1 Let us then
put
1198831199051
= 119868 [1198801199051
le 119909] minus 119909 1198831199052
= 119868 [1198801199052
le 119909] minus 119909 (50)
We will now estimate the summands in (48)If max119894 119895 119896 = 119894 then
10038161003816100381610038161003816119864 119883
119905sdot (119883
119905+119894119883
119905+119894+119895119883
119905+119894+119895+119896)10038161003816100381610038161003816
=100381610038161003816100381610038161003816Cov (119883
119905 119883
119905+119894119883
119905+119894+119895119883
119905+119894+119895+119896)100381610038161003816100381610038161003816le 119862
1198944
(51)
Similarly for max119894 119895 119896 = 119896 we have
10038161003816100381610038161003816119864 (119883
119905119883
119905+119894119883
119905+119894+119895) sdot 119883
119905+119894+119895+11989610038161003816100381610038161003816
=10038161003816100381610038161003816Cov (119883
119905119883
119905+119894119883
119905+119894+119895 119883
119905+119894+119895+119896)10038161003816100381610038161003816le 119862
1198964
(52)
When max119894 119895 119896 = 119895 then
10038161003816100381610038161003816119864 (119883
119905119883
119905+119894) sdot (119883
119905+119894+119895119883
119905+119894+119895+119896)10038161003816100381610038161003816
=10038161003816100381610038161003816Cov (119883
119905119883
119905+119894 119883
119905+119894+119895119883
119905+119894+119895+119896)
+119864 (119883119905119883
119905+119894) 119864 (119883
119905+119894+119895119883
119905+119894+119895+119896)10038161003816100381610038161003816
le 1198621198954
+ 1198621198942119862
1198962
(53)
We keep on estimating the fourth moment of 119878119899by intro-
ducing 119905 isin 0 119899mdashthe index for which |119864(119883119905119883
119905+119894119883
119905+119894+119895
119883119905+119894+119895+119896
)| attains its maximum
1198641198784
119899le 4119899 sum
0le119894119895119896le119899minus1
10038161003816100381610038161003816119864 (119883
119905119883
119905+119894119883
119905+119894+119895119883
119905+119894+119895+119896)10038161003816100381610038161003816
=4119899( sum
0le119894119896le119895
[1198621198942sdot 119862
1198962+ 119862
1198954]+ sum
0le119894119895le119896
1198621198964+ sum
0le119896119895le119894
1198621198944)
= 4119899( sum
0le119894119896le119895
1198621198942sdot 119862
1198962+ 3 sum
0le119896119895le119894
1198621198944)
(54)
6 ISRN Probability and Statistics
The terms obtained in (54) may further be bounded fromabove in the following way
sum
0le119894119896le119895
1198621198942sdot 119862
1198962=
119899minus1
sum
119895=0
119895
sum
119894119896=0
1198621198942119862
1198962
le 119899
119899minus1
sum
119894119896=0
1198621198942119862
1198962= 119899(
119899minus1
sum
119894=0
1198621198942)
2
sum
0le119896119895le119894
1198621198944
=
119899minus1
sum
119894=0
119894
sum
119895119896=0
1198621198944
le
119899minus1
sum
119894=0
(119894 + 1) (119894 + 1) 1198621198944
=
119899minus1
sum
119894=0
(119894 + 1)2119862
1198944
(55)
We thus get the inequality
1198641198784
119899le 4([119899
119899minus1
sum
119894=0
1198621198942]
2
+ 3119899
119899minus1
sum
119894=0
(119894 + 1)2119862
1198944) (56)
We shall now focus on estimating 1198621199032
and 1198621199034 Since 119880
1199051
and 1198801199052
are associated uniformly [0 1] distributed rvrsquos 1198831199051
and 1198831199052
mdashas monotone functions of these rvrsquosmdashare associat-ed as well In order to bound
1198621199032
= sup0le1199051lt1199052le119899 1199052minus1199051=119903
Cov (1198831199051
1198831199052
) (57)
let us notice that from Schwarz inequality
Cov (1198831199051
1198831199052
) = 119864 (119868 [1198801199051
le 119909] 119868 [1198801199052
le 119909]) minus 1199092
le radic1199092 minus 1199092le 119909
(58)
On the other hand invoking inequalities from [13 15]
Cov (1198831199051
1198831199052
)
le sup119909isin[01]
[119875 (1198801199051
le 1199091198801199052
le 119909) minus 119875 (1198801199051
le 119909)119875 (1198801199052
le 119909)]
le 119862 sdot Cov13
(1198801199051
1198801199052
) for associated rvrsquos4 sdot Cov (119880
1199051
1198801199052
) for MTP2rvrsquos
(59)
As a consequence
1198621199032
le min 119909 119862 sdot Cov13
(1198801199051
1198801199052
) for associated rvrsquosmin 119909 4 sdot Cov (119880
1199051
1198801199052
) for MTP2rvrsquos
(60)
where 1199052minus 119905
1= 119903 Still we need the upper bound for 119862
1199034 It
turns out that119862
1199034
= sup1003816100381610038161003816100381610038161003816100381610038161003816
Cov2
prod
119894=1
(119868 [119880119905119894
le 119909] minus 119909)
4
prod
119894=3
(119868 [119880119905119894
le 119909] minus 119909)
1003816100381610038161003816100381610038161003816100381610038161003816
(61)
In other words among all divisions of the group 119883119905119894
119894 isin
1 2 3 4 into two subgroups 1198621199034
attains the biggest valuein case we take two subgroups consisted of two rvrsquos Thesupremum runs over the set 119905
1 119905
2 119905
3 119905
4isin N 0 le 119905
1lt 119905
2lt
1199053lt 119905
4le 119899 and 119905
3minus 119905
2= 119903 Elementary calculation leads to the
following formula
1198621199034
= sup 10038161003816100381610038161003816119875 (119880119905119894
le 119909 119894 isin 1 2 3 4)
minus 119875 (119880119905119894
le 119909 119894 isin 1 2) 119875 (119880119905119894
le 119909 119894 isin 3 4)
minus 119909 [119875 (119880119905119894
le 119909 119894 isin 1 2 3)
minus 119875 (119880119905119894
le 119909 119894 isin 1 2) 119875 (1198801199053
le 119909)
+ 119875 (119880119905119894
le 119909 119894 isin 1 3 4)
minus 119875 (119880119905119894
le 119909 119894 isin 3 4) 119875 (1198801199051
le 119909)
+ 119875 (119880119905119894
le 119909 119894 isin 1 2 4)
minus 119875 (119880119905119894
le 119909 119894 isin 1 2) 119875 (1198801199054
le 119909)
+ 119875 (119880119905119894
le 119909 119894 isin 2 3 4)
minus119875 (119880119905119894
le 119909 119894 isin 3 4) 119875 (1198801199052
le 119909)]
+ 1199092[119875 (119880
1199051
le 1199091198801199053
le 119909)
minus 119875 (1198801199051
le 119909)119875 (1198801199053
le 119909)
+ 119875 (1198801199051
le 1199091198801199054
le 119909)
minus 119875 (1198801199051
le 119909)119875 (1198801199054
le 119909)
+ 119875 (1198801199052
le 1199091198801199053
le 119909)
minus 119875 (1198801199052
le 119909)119875 (1198801199053
le 119909)
+ 119875 (1198801199052
le 1199091198801199054
le 119909)
minus119875 (1198801199052
le 119909)119875 (1198801199052
le 119909)]10038161003816100381610038161003816
le10038161003816100381610038161198771
1003816100381610038161003816 +10038161003816100381610038161198772
1003816100381610038161003816 +10038161003816100381610038161198773
1003816100381610038161003816
(62)
where for the sake of simplicity 1198771 119877
2 and 119877
3are respec-
tively the free coefficient the expression with 119909 and theexpression with 119909
2 Using the Lebowitz inequality (see [15]for instance) and inequalities obtained in [13 15] we arrive at10038161003816100381610038161198771
1003816100381610038161003816 =10038161003816100381610038161003816119875 (119880
119905119894
le 119909 119894 isin 1 2 3 4)
minus119875 (119880119905119894
le 119909 119894 isin 1 2) 119875 (119880119905119894
le 119909 119894 isin 3 4)10038161003816100381610038161003816
le 11986711988011990511198801199053
+ 11986711988011990511198801199054
+ 11986711988011990521198801199053
+ 11986711988011990521198801199054
ISRN Probability and Statistics 7
le
119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
)) for associated rvrsquos
4(
2
sum
119894=1
4
sum
119895=3
Cov (119880119905119894
119880119905119895
)) for MTP2rvrsquos
(63)
Analogously we get
10038161003816100381610038161198772
1003816100381610038161003816
le
2119909 sdot 119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
)) for associated rvrsquos
2119909 sdot 4(
2
sum
119894=1
4
sum
119895=3
Cov (119880119905119894
119880119905119895
)) for MTP2rvrsquos
10038161003816100381610038161198773
1003816100381610038161003816
le
1199092sdot 119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
) ) for associated rvrsquos
1199092sdot 4(
2
sum
119894=1
4
sum
119895=3
Cov (119880119905119894
119880119905119895
)) for MTP2rvrsquos
(64)
Eventually we have
1198621199034
le
119891 (119909) sdot119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
)) for associated rvrsquos
119891 (119909) sdot4(
2
sum
119894=1
4
sum
119895=3
Cov (119880119905119894
119880119905119895
)) for MTP2rvrsquos
(65)
where 1199053minus 119905
2= 119903 and 119891(119909) = (1 + 2119909 + 119909
2) for 119909 isin [0 1]
Let us now introduce the following notation
120579119903=
sup119905isinN
Cov (119880119905 119880
119905+119903) for 119903 isin 1 2
1 for 119903 = 0
(66)
and assume it decays powerly at rate 119886 in the following way
120579119903= 119874(
1
(119903 + 1)119886) for 119886 gt 0 119903 isin 0 1 (67)
Let us get back to inequality (56) we can now carry on Atfirst for associated rvrsquos
1198641198784
119899le 4[119899
119899minus1
sum
119903=0
min 119909 119862 sdot Cov13(119880
1199051
1198801199052
)]
2
+ 4 sdot 3119899
119899minus1
sum
119903=0
(119903 + 1)2119891 (119909) sdot 119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
))
le 119863 sdot ([119899
119899minus1
sum
119903=0
min 119909 (119903 + 1)minus1198863
]
2
+119899
119899minus1
sum
119903=0
(119903 + 1)2(119903 + 1)
minus1198863)
= 119863 sdot ([119899
119899
sum
119903=1
min 119909 119903minus1198863]
2
+ 119899
119899
sum
119903=1
1199032119903minus1198863
)
le 119863 sdot ([
[
119899 sum
119903lt119909minus3119886
119909 + 119899 sum
119903ge119909minus3119886
1
1199031198863]
]
2
+ 119899
119899
sum
119903=1
1199032minus1198863
)
le 1198631sdot (119899
2119909
2(119886minus3)119886+ 120585)
(68)
where119863 and1198631are constants and
120585 = 119899
119899
sum
119903=1
1199032minus1198863
=
119874(119899) 119886 gt 9
119874 (119899 ln 119899) 119886 = 9
119874 (1198994minus1198863
) 3 lt 119886 lt 9
(69)
It is worth mentioning that in the last inequality of (68) weused the estimate
int
infin
119909
1
119905119901119889119905 sim
1
119909119901minus1for 119901 gt 1 (70)
At the same time in the case of MTP2rvrsquos we get
1198641198784
119899le 119863
2sdot (119899
2119909
2(119886minus1)119886+ 120577) (71)
where1198632is constant and
120577 = 119899
119899
sum
119903=1
1199032minus119886
=
119874(119899) 119886 gt 3
119874 (119899 ln 119899) 119886 = 3
119874 (1198994minus119886
) 1 lt 119886 lt 3
(72)
Let now 120572119899(119909) = (1radic119899)sum
119899
119894=1(119868[119880
119905119894
le 119909] minus 119909) Then
120572119899(119909) minus 120572
119899(119910) =
1
radic119899
119899
sum
119894=1
(119868 [119909 lt 119880119905119894
le 119910] minus (119909 minus 119910))
for 119909 119910 isin [0 1]
(73)
8 ISRN Probability and Statistics
has the fourth moment estimatedmdashin the case of associatedrvrsquosmdashby
119864[120572119899(119909) minus 120572
119899(119910)]
4
= 119864[1
radic119899
119899
sum
119894=1
(119868 [119909 lt 119880119905119894
le 119910] minus (119909 minus 119910))]
4
=1
1198992119864[
119899
sum
119894=1
(119868 [119909 lt 119880119905119894
le 119910] minus (119909 minus 119910))]
4
le 1198631sdot (
1
119899211989921003816100381610038161003816119909 minus 119910
1003816100381610038161003816
2(119886minus3)119886
+120585
1198992)
= 1198631sdot (
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2(119886minus3)119886
+120585
1198992)
=
119874(119899minus1) 119886 gt 9
119874(ln 119899119899
) 119886 = 9
119874 (1198992minus1198863
) 3 lt 119886 lt 9
(74)
and in the case of MTP2rvrsquos by
119864[120572119899(119909) minus 120572
119899(119910)]
4
le 1198632sdot (
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2(119886minus1)119886
+120577
1198992)
=
119874(119899minus1) 119886 gt 3
119874(ln 119899119899
) 119886 = 3
119874 (1198992minus119886
) 1 lt 119886 lt 3
(75)
In light of the Shao and Yursquos criterion our process is tight forassociated rvrsquos when 119886 gt 6 and for MTP
2rvrsquos when 119886 gt 2
Let us sum up this result in the following theorem
Theorem 3 Let 120572119899(119909) = (1radic119899)sum
119899
119894=1(119868[119880
119894le 119909] minus 119909) be the
empirical process built on an associated sequence of uniformly[0 1] distributed rvrsquos 119880
119894119894ge1
Let also
120579119903= sup
119905isinNCov (119880
119905 119880
119905+119903) = 119874 ((119903 + 1)
minus119886)
where 119903 isin 0 1 119886 gt 0
(76)
Then 120572119899(119909)
119899ge1is tight for 119886 gt 6 If the rvrsquos 119880
119894119894ge1
are MTP2
then the process is tight for 119886 gt 2
Yu assumed stationarity of 119880119894119894ge1
andsum
infin
119899=111989965+]Cov(119880
0 119880
119899) lt infin for a positive constant ]
thus the rate of decay 119886 gt 7 5 Our result weakens consid-erably these assumptions especially in the case of MTP
2rvrsquos
Louhichi in [16] proposed a different tightness criterioninvolving the so-called bracketing numbers She managedto enhance Yursquos resultmdasheven more than Shao and Yu in[8]mdashsince she proved that it suffices to take 119886 gt 4 to gettightness of the empirical process based on the associated
rvrsquos Nevertheless she kept the assumption of stationarityvalid
In the final analysis our resultrsquos advantage is the absenceof the stationarity assumption and the rate of decay for120579119903remains (up to the authorrsquos knowledge) unimproved for
MTP2rvrsquos
Unfortunately with a view to obtainingweak convergenceof the process in question that is also convergence offinite-dimensional distributions we do not know how tomanage without the assumption of stationarityTherefore weconclude with the following corollary
Corollary 4 Let 120572119899(119909) = (1radic119899)sum
119899
119894=1(119868[119880
119894le 119909] minus 119909) be the
empirical process built on a stationary associated sequence ofuniformly [0 1] distributed rvrsquos 119880
119894119894ge1
Let also
120579119903= Cov (119880
1 119880
1+119903) = 119874 ((119903 + 1)
minus119886)
where 119903 isin 0 1 119886 gt 0
(77)
Then if 119886 gt 6
120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (78)
where 119861(sdot) is the zero mean Gaussian process on [0 1] withcovariance structure defined by
1205902(119909 119910) = 119909 and 119910 minus 119909119910
+
infin
sum
119895=1
[ Cov (119868 [1198801le 119909] 119868 [119880
119895+1le 119910])
+ Cov (119868 [1198801le 119910] 119868 [119880
119895+1le 119909])]
(79)
If the rvrsquos 119880119894119894ge1
are MTP2 then it suffices to be 119886 gt 2 in
order to claim the above convergence
Proof It remains to establish convergence of finite-dimen-sional distributions repeating the procedure from [7]
4 Tightness of the Kernel-TypeEmpirical Process
In this section we shall weaken assumption imposed on thecovariance structure of rvrsquos 119883
119895119895ge1
by Cai and Roussas in [1]for the kernel estimator of the df
119865119899(119909) =
1
119899
119899
sum
119895=1
119870(
119909 minus 119883119895
ℎ119899
) (80)
They deal with a stationary sequence of negatively associatedrvrsquos (cf [17]) and need the same condition as Yu [7] that is
1003816100381610038161003816Cov (1198831 119883
1+119899)1003816100381610038161003816 = 119874(
1
11989975+]) (81)
to get tightness of the smooth empirical process (see condi-tion (A4) in [1])
ISRN Probability and Statistics 9
It turns out that it suffices to have
1003816100381610038161003816Cov (1198801 119880
1+119899)1003816100381610038161003816 = 119874(
1
1198993119901(119901minus1)) (82)
where 119901 gt 4 is a positive constant taken from the tightnesscriterion (8) It is easy to see that asymptotically we get therate 3
On the way to prove it we will also take use of aRosenthal-type inequality due to Shao andYu (seeTheorem 2in [8]) we shall recall in the following lemma
Lemma 5 Let 119901 gt 2 and 119891 be a real valued function boundedby 1 with bounded first derivative Suppose that 119883
119899119899ge1
is asequence of stationary and associated rvrsquos such that for 119899 isin N
Cov (1198831 119883
119899) = 119874 (119899
minus119887) for some 119887 gt 119901 minus 1 (83)
Then for any 120583 gt 0 there exists some positive constant 119896120583
independent of the function 119891 for which
1198641003816100381610038161003816119878119899
(119891) minus 119864119878119899(119891)
1003816100381610038161003816
119901
le 119896120583(119899
1+12058310038171003817100381710038171003817119891
101584010038171003817100381710038171003817
2
+1198991199012 [
[
119899
sum
119895=1
10038161003816100381610038161003816Cov (119891 (119883
1)119891 (119883
119895))10038161003816100381610038161003816]
]
)
(84)
As we can see the lemma assumes association but itworks for negatively associated rvrsquos as well since in the proofit reaches back the result of Newman (see Proposition 15 in[18]) where both types of association are allowed
Let us recall that 120572119899(119909) = radic119899(119865
119899(119909) minus 119864119865
119899(119909)) where
119865119899(119909) = (1119899)sum
119899
119895=1119870((119909 minus 119883
119895)ℎ
119899) It is easy to see that
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
= 119899minus1199012
11986410038161003816100381610038161003816119878119899(119891 (119883
119895)) minus 119864119878
119899(119891 (119883
119895))10038161003816100381610038161003816
119901
(85)
where
119891 (119883119895) = 119870(
119909 minus 119883119895
ℎ119899
) minus 119870(
119910 minus 119883119895
ℎ119899
)
119878119899(119891) =
119899
sum
119895=1
119891 (119883119895)
(86)
With an intent to use Lemma 5 we need 119891 to be boundedfrom above by 1 (which in our case is obvious) and to havebounded 119891
1015840 thus we assume
sup119905isinR
10038161003816100381610038161003816119870
1015840(119905)
10038161003816100381610038161003816
1
ℎ119899
le119862
119870
ℎ119899
lt infin (87)
Now applying inequality (84) we have
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
le119896120583119899minus1199012
1198991+120583
1198622
119870
ℎ2
119899
+1198991199012[
[
119899
sum
119895=1
10038161003816100381610038161003816Cov (119891 (119883
1)119891 (119883
119895))10038161003816100381610038161003816]
]
1199012
(88)
Newman showed in [18] that
Cov (1198911(119883) 119891
2(119884))
= ∬R2119891
1015840
1(119909) 119891
1015840
2(119910)
times [119875 (119883 le 119909 119884 le 119910) minus 119875 (119883 le 119909) 119875 (119884 le 119910)] 119889119909 119889119910
(89)
if 1198911and 119891
2are real valued functions on R having square
integrable derivatives 1198911015840
1and 119891
1015840
2 respectively and provided
that 1198911(119883) 119891
2(119884) have finite secondmoments In light of that
equation and linearity of covariance we get10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
=
1003816100381610038161003816100381610038161003816∬
R2[119875 (119883
1lt 119909 minus ℎ
119899119904 119883
119895lt 119909 minus ℎ
119899119905)
minus119875 (1198831lt 119909 minus ℎ
119899119904 119883
119895lt 119910 minus ℎ
119899119905)]
minus [119875 (1198831lt 119909 minus ℎ
119899119904) 119875 (119883
119895lt 119909 minus ℎ
119899119905)
minus119875 (1198831lt 119909 minus ℎ
119899119904) 119875 (119883
119895lt 119910 minus ℎ
119899119905)]
minus [119875 (1198831lt 119910 minus ℎ
119899119904 119883
119895lt 119909 minus ℎ
119899119905)
minus119875 (1198831lt 119910 minus ℎ
119899119904 119883
119895lt 119910 minus ℎ
119899119905)]
minus [119875 (1198831lt 119910 minus ℎ
119899119904) 119875 (119883
119895lt 119910 minus ℎ
119899119905)
minus 119875 (1198831lt 119910 minus ℎ
119899119904)
times 119875 (119883119895lt 119909 minus ℎ
119899119905)] 119896 (119904) 119896 (119905) 119889119904 119889119905
10038161003816100381610038161003816100381610038161003816
(90)
Without the loss of generality we may and do assume that119909 gt 119910 thus10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
=
1003816100381610038161003816100381610038161003816∬
R2119875 (119910 minus ℎ
119899119904 le 119883
1lt 119909 minus ℎ
119899119904
119910 minus ℎ119899119905 le 119883
119895lt 119909 minus ℎ
119899119905)
minus 119875 (119910 minus ℎ119899119904 le 119883
1lt 119909 minus ℎ
119899119904)
times 119875 (119910 minus ℎ119899119905 le 119883
119895lt 119909 minus ℎ
119899119905) 119896 (119904) 119896 (119905) 119889119904 119889119905
1003816100381610038161003816100381610038161003816
(91)
and by triangle inequality we get10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
le ∬R2[int
119909minusℎ119899119904
119910minusℎ119899119904
int
119909minusℎ119899119905
119910minusℎ119899119905
1198911198831119883119895(119906 V) 119889119906 119889V
+int
119909minusℎ119899119904
119910minusℎ119899119904
1198911198831(119906) 119889119906int
119909minusℎ119899119905
119910minusℎ119899119905
119891119883119895(V) 119889V] 119896 (119904) 119896 (119905) 119889119904 119889119905
(92)
10 ISRN Probability and Statistics
1198911198831119883119895
(119906 V) 1198911198831
(119906) and 119891119883119895
(V) are the joint pdf of [1198831 119883
119895]
and marginal pdfrsquos of rvrsquos1198831and119883
119895 respectively We need
to further assume that 1198621stands for a common upper bound
of 1198911198831
and 1198911198831119883119895
for all 119895 isin N that is
100381710038171003817100381710038171198911198831
10038171003817100381710038171003817le 119862
1
1003817100381710038171003817100381710038171198911198831119883119895
100381710038171003817100381710038171003817le 119862
1forall119895 ge 1 (93)
Then we finally obtain
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816le 2119862
1
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
where = max 1198621 119862
2
1
(94)
On the other hand proceeding like Cai and Roussas in [1] wecan shortly get
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
le
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119909 minus 119883
1
ℎ119899
) 119870(
119909 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119909 minus 119883
1
ℎ119899
) 119870(
119910 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119910 minus 119883
1
ℎ119899
) 119870(
119909 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119910 minus 119883
1
ℎ119899
) 119870(
119910 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
le 41198622
10038161003816100381610038161003816Cov (119883
1 119883
119895)10038161003816100381610038161003816
13
(95)
where 1198622is a constant relevant to the covariance inequality
for negatively associated rvrsquos (see [13])We now arrive at the following inequality
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
le max 1198621 119862
2
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
10038161003816100381610038161003816Cov(119883
1 119883
119895)10038161003816100381610038161003816
13
(96)
Assuming |Cov(119891(1198831) 119891(119883
119895))| = 119874(119895
minus119903) and proceeding
similarly to (68) we can get
[
[
119899
sum
119895=1
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816]
]
1199012
le max 1198621 119862
2 [
[
sum
119895lt|119909minus119910|minus2(1199033)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
+ sum
119895ge|119909minus119910|minus2(1199033)
1
1198951199033]
]
1199012
le 2max 1198621 119862
21003816100381610038161003816119909 minus 119910
1003816100381610038161003816
((119903minus3)119903)119901
(97)
To sum up we obtain the following inequality
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
le 119896120583119899
1+120583minus1199012119862
2
119870
ℎ2
119899
+ 2max 1198621 119862
21003816100381610038161003816119909 minus 119910
1003816100381610038161003816
((119903minus3)119903)119901
le 1198621198991+120583minus1199012 1
ℎ2
119899
+1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
((119903minus3)119903)119901
(98)
where 119862 = 119896120583max1198622
119896 2max119862
1 119862
2 The formula of the
tightness criterion (8) implies that ((119903 minus 3)119903)119901 gt 1 so
119903 gt3119901
119901 minus 1 where 119901 gt 2 (99)
Let us recall the assumptions imposed on the bandwidhtsℎ
119899119899ge1
by Cai and Roussas in [1]B1 lim
119899rarrinfinℎ119899= 0 and ℎ
119899gt 0 for all 119899 isin N
+
B2 lim119899rarrinfin
119899ℎ119899= infin thus ℎ
119899= 119874(1119899
1minus120573) 120573 gt 0
B3 lim119899rarrinfin
119899ℎ4
119899= 0 hence ℎ
119899= 119874(1119899
14+120575) 120575 gt 0
In light of the above
1198991+120583minus1199012 1
ℎ2
119899
= 11989932+120583+2120575minus1199012
(100)
where 120583 gt 0 119901 gt 2 and 0 lt 120575 lt 34 Confrontation with thetightness criterion (8) forces
3
2+ 120583 + 2120575 minus
119901
2lt minus
1
2 (101)
which implies 119901 gt 4 Let us conclude with the followingtheorem
Theorem6 Let 120572119899(119909) = radic119899(119865
119899(119909)minus119864119865
119899(119909)) be the empirical
process built on the kernel estimator of the df119865 for a stationarysequence of negatively associated rvrsquos Assume that conditionsA1 A2 (87) B1 B2 B3 are satisfied Then provided that thetightness criterion (8) holds with 119901 gt 4 it suffices to demand
1003816100381610038161003816Cov (1198801 119880
1+119899)1003816100381610038161003816 = 119874(
1
1198993119901(119901minus1)) (102)
in order to obtain120572
119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (103)
where 119861(sdot) is the zero mean Gaussian process with covariancestructure defined by
1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)
+
infin
sum
119895=1
[ Cov (119868 [1198831le 119909] 119868 [119883
119895+1le 119910])
+ Cov (119868 [1198831le 119910] 119868 [119883
119895+1le 119909])]
(104)
and 119909 119910 isin [0 1]
Proof The remaining covergence of finite-dimensional dis-tributions of 120572
119899(119909) is established in [1]
ISRN Probability and Statistics 11
5 Weak Convergence of theRecursive Kernel-Type Empirical Processunder IID Assumption
The aim of this section is to show weak convergence of theempirical process
120572119899(119909) =
1
radic119899
119899
sum
119895=1
[119870(
119909 minus 119883119895
ℎ119895
) minus 119864119870(
119909 minus 119883119895
ℎ119895
)] (105)
built on iid rvrsquos 119883119895119895ge1
Wewill prove that120572
119899(sdot) convergesweakly to the Brownian
bridge 119861(sdot) with the following covariance structure
1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910) for 119909 119910 isin [0 1]
(106)
It turns out that the tightness criterion (8) suffices to reachthe goal Convergence of finite-dimensional distributions of120572
119899(sdot) holds and we shall show itProceeding like Cai and Roussas in [1] we need to show
that for any 119886 119887 isin R
119886120572119899(119909) + 119887120572
119899(119910) 997888rarr 119886119861 (119909) + 119887119861 (119910) in distribution
(107)
Let us introduce the notation
119884119894(119909) = 119870(
119909 minus 119883119894
ℎ119894
) minus 119864119870(119909 minus 119883
119894
ℎ119894
) (108)
and look into the covariance structure of 120572119899(sdot)
Cov (120572119899(119909) 120572
119899(119910))
= Cov( 1
radic119899
119899
sum
119894=1
119884119894(119909)
1
radic119899
119899
sum
119894=1
119884119894(119910))
=1
119899
119899
sum
119894=1
Cov (119884119894(119909) 119884
119894(119910))
=1
119899
119899
sum
119894=1
Cov(119870(119909 minus 119883
119894
ℎ119894
) 119870(119910 minus 119883
119894
ℎ119894
))
(109)
where the second equality follows from assumed indepen-dence of rvrsquos 119883
119894119894ge1
Firstly we observe that each summand converges to119865(119909and
119910) minus 119865(119909)119865(119910) since
Cov(119870(119909 minus 119883
119894
ℎ119894
) 119870(119910 minus 119883
119894
ℎ119894
))
= 119864[119870(119909 minus 119883
119894
ℎ119894
)119870(119910 minus 119883
119894
ℎ119894
)]
minus 119864119870(119909 minus 119883
119894
ℎ119894
)119864119870(119910 minus 119883
119894
ℎ119894
)
(110)
where
119864[119870(119909 minus 119883
119894
ℎ119894
)119870(119910 minus 119883
119894
ℎ119894
)]
= int
119909and119910
minusinfin
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
+ int
119909or119910
119909and119910
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
+ int
infin
119909or119910
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
(111)
and 119865 denotes the common df of the rvrsquos 119883119894119894ge1
In [6] itwas shown that
119864119870(119909 minus 119883
119894
ℎ119894
) 997888rarr 119865 (119909) (112)
and recalling that the kernel function 119870 is a df as well wearrive at the conclusion
Secondly applying Toeplitz lemma we get
Cov (120572119899(119909) 120572
119899(119910)) 997888rarr 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)
= 1205902(119909 119910) 119899 997888rarr infin
(113)
Thus
Var (119886120572119899(119909) + 119887120572
119899(119910))
= 1198862 Var (120572
119899(119909)) + 2119886119887Cov (120572
119899(119909) 120572
119899(119910))
+ 1198872 Var (120572
119899(119910))
997888rarr 1198862120590
2(119909 119909) + 2119886119887120590
2(119909 119910) + 119887
2120590
2(119910 119910) 119899 997888rarr infin
(114)
We are now on the way to prove that 119886120572119899(119909) + 119887120572
119899(119910) con-
verges in distribution to 119886119861(119909) + 119887119861(119910) sim N(0 1198862120590
2(119909 119909) +
21198861198871205902(119909 119910) + 119887
2120590
2(119910 119910)) which in other words means that
1
radic119899
infin
sum
119894=1
119886119884119894(119909) + 119887119884
119894(119910)
radicVar (119886120572119899(119909) + 119887120572
119899(119910))
997888rarr N (0 1) (115)
In order to obtain (115) we will use Lyapunov condition
lim119899rarrinfin
1
1198632+120575
119899
119899
sum
119894=1
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884119894(119910)
1003816100381610038161003816
2+120575
= 0 (116)
for 120575 = 1 where 1198632
119899= Var(sum119899
119894=1[119886119884
119894(119909) + 119887119884
119894(119910)]) Since
|119870(sdot) minus 119864119870(sdot)| le 2
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884i (119910)1003816100381610038161003816
3
le 8 (1198863+ 119887
3) (117)
Moreover 1198632
119899= 119899Var(119886120572
119899(119909) + 119887120572
119899(119910)) together with (114)
yields
1198633
119899= 119874 (119899
32) (118)
12 ISRN Probability and Statistics
Combining (117) with (118) we obtain
1
1198633
119899
119899
sum
119894=1
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884119894(119910)
1003816100381610038161003816
3
= 119874 (119899minus12
) (119)
which shows that Lyapunov condition holds and completesthe proof of convergence of finite-dimensional distributionsof 120572
119899(119909)We summarize the result of that section in the following
theorem
Theorem 7 Let 120572119899(119909) be the recursive kernel-type empirical
process defined by (105) built on iid rvrsquos 119883119895119895ge1
withmarginal df 119865 If 120572
119899(sdot) satisfies the tightness criterion (8) then
120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (120)
where 119861 is the Brownian bridge
Acknowledgment
The author would like to thank the referees for the carefulreading of the paper and helpful remarks
References
[1] Z Cai and G G Roussas ldquoWeak convergence for smoothestimator of a distribution function under negative associationrdquoStochastic Analysis and Applications vol 17 no 2 pp 145ndash1681999
[2] Z Cai andG G Roussas ldquoEfficient Estimation of a DistributionFunction under Qudrant Dependencerdquo Scandinavian Journal ofStatistics vol 25 no 1 pp 211ndash224 1998
[3] ZW Cai and G G Roussas ldquoUniform strong estimation under120572-mixing with ratesrdquo Statistics amp Probability Letters vol 15 no1 pp 47ndash55 1992
[4] G G Roussas ldquoKernel estimates under association stronguniform consistencyrdquo Statistics amp Probability Letters vol 12 no5 pp 393ndash403 1991
[5] Y F Li and S C Yang ldquoUniformly asymptotic normalityof the smooth estimation of the distribution function underassociated samplesrdquo Acta Mathematicae Applicatae Sinica vol28 no 4 pp 639ndash651 2005
[6] Y Li C Wei and S Yang ldquoThe recursive kernel distributionfunction estimator based onnegatively and positively associatedsequencesrdquo Communications in Statistics vol 39 no 20 pp3585ndash3595 2010
[7] H Yu ldquoA Glivenko-Cantelli lemma and weak convergence forempirical processes of associated sequencesrdquo ProbabilityTheoryand Related Fields vol 95 no 3 pp 357ndash370 1993
[8] Q M Shao and H Yu ldquoWeak convergence for weightedempirical processes of dependent sequencesrdquo The Annals ofProbability vol 24 no 4 pp 2098ndash2127 1996
[9] P Billingsley Convergence of Probability Measures Wiley Seriesin Probability and Statistics Probability and Statistics JohnWiley amp Sons New York NY USA 1999
[10] J D Esary F Proschan and D W Walkup ldquoAssociation ofrandom variables with applicationsrdquo Annals of MathematicalStatistics vol 38 pp 1466ndash1474 1967
[11] A Bulinski and A Shashkin Limit Theorems for AssociatedRandom Fields and Related Systems vol 10 of Advanced Serieson Statistical Science and Applied Probability World Scientific2007
[12] S Karlin and Y Rinott ldquoClasses of orderings of measures andrelated correlation inequalities I Multivariate totally positivedistributionsrdquo Journal of Multivariate Analysis vol 10 no 4 pp467ndash498 1980
[13] P Matuła and M Ziemba ldquoGeneralized covariance inequali-tiesrdquo Central European Journal of Mathematics vol 9 no 2 pp281ndash293 2011
[14] P Doukhan and S Louhichi ldquoA new weak dependence condi-tion and applications to moment inequalitiesrdquo Stochastic Proc-esses and their Applications vol 84 no 2 pp 313ndash342 1999
[15] P Matuła ldquoA note on some inequalities for certain classes ofpositively dependent random variablesrdquo Probability and Math-ematical Statistics vol 24 no 1 pp 17ndash26 2004
[16] S Louhichi ldquoWeak convergence for empirical processes of asso-ciated sequencesrdquo Annales de lrsquoInstitut Henri Poincare vol 36no 5 pp 547ndash567 2000
[17] K Joag-Dev and F Proschan ldquoNegative association of randomvariables with applicationsrdquoThe Annals of Statistics vol 11 no1 pp 286ndash295 1983
[18] C M Newman ldquoAsymptotic independence and limit theoremsfor positively and negatively dependent random variablesrdquo inInequalities in Statistics and Probability Y L Tong Ed vol 5pp 127ndash140 Institute ofMathematical StatisticsHayward CalifUSA 1984
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
Volume 2014
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
ISRN Probability and Statistics 5
= 119875( max1le119894le2(119898
119899+1)
1003816100381610038161003816120572119899(119910 + 119894119903
119899) minus 120572
119899(119910)
1003816100381610038161003816 ge 119872120598)
le
119862119901min119901
11199012+1199013
(119872120598)119901
(2120575)min119901
11199012+1199013le 120578
(43)
for any fixed 119910 isin [0 1] Since the upper bound does notdepend on 119910 and the probability measure as well as supre-mum function are continuous we obtain
119875( sup0le119910le1minus2120575
sup119910le119909le119910+2120575
1003816100381610038161003816120572119899(119909) minus 120572
119899(119910)
1003816100381610038161003816 ge 4119872120598 + 119862radic119899ℎ4
119899) le 120578
(44)
where 4119872120598 + 119862radic119899ℎ4
119899is arbitrarily small
Since condition (10) is checked the proof is completed
3 Tightness of the Standard Empirical Process
In this section we deal with the standard empirical processbuilt on an associated sequence of uniformly [0 1]distributedrvrsquos 119880
119895119895ge1
that is
120572119899(119909) =
1
radic119899
119899
sum
119895=1
(119868 [119880119895le 119909] minus 119909) (45)
where 119909 isin [0 1] We shall relax the restrictions imposedon the process by Yu in [7] to obtain tightness Preciselywe do not need stationarity any more due to the techniquedrawn from [14] and we lower the assumed rate at which thecovariance tends to zero
While proving tightness of the empirical process we willuse the criterion proved in the first section as well as some ofour covariance inequalities
We shall start with the fact known under the name ofmultinomial theorem We recall it in the following lemma
Lemma 2 For natural numbers119898 119899 and 1199091 119909
2 119909
119898isin R
(1199091+ sdot sdot sdot + 119909
119899)119898
= sum
1198961+sdotsdotsdot+119896
119899=119898
(119898
1198961 119896
119899
) prod
1le119894le119899
119909119896119894
119894 (46)
where (119898
1198961 119896
119899) = 119898(119896
1 sdot sdot sdot 119896
119899) and 119896
1 119896
119899isin
0 1 119898
In particular
(1199091+ sdot sdot sdot + 119909
119899)4
le 4 sdot sum
1198961+sdotsdotsdot+119896
119899=4
1199091198961
1sdot sdot sdot 119909
119896119899
119899 (47)
which implies for 119878119899= sum
119899
119894=1119883
119894 that
1198641198784
119899le 4 sdot sum
1198961+sdotsdotsdot+119896
119899=4
119864 (1198831198961
1sdot sdot sdot 119883
119896119899
119899)
le 4 sdot
119899
sum
119905=1
sum
0le119894119895119896le119899minus1
10038161003816100381610038161003816119864 (119883
119905119883
119905+119894119883
119905+119894+119895119883
119905+119894+119895+119896)10038161003816100381610038161003816
(48)
Let us now introduce the following notation due to Doukhanand Louhichi (see [14]) for a sequence of centered rvrsquos119883
119905119899
119899ge1
119862119903119902
= sup100381610038161003816100381610038161003816Cov (119883
1199051
sdot sdot sdot 119883119905119898
119883119905119898+119903sdot sdot sdot 119883
119905119902
)100381610038161003816100381610038161003816 (49)
where the supremum runs over all divisions of the groupcomposed of 119902 rvrsquos into two subgroups such that the distancebetween the highest index of the rvrsquos in the first group and thelowest index of the rvrsquos from the second group is equal to 119903119903 isin 1 2 For 119903 = 0 we shall define 119862
0119902= 1 Let us then
put
1198831199051
= 119868 [1198801199051
le 119909] minus 119909 1198831199052
= 119868 [1198801199052
le 119909] minus 119909 (50)
We will now estimate the summands in (48)If max119894 119895 119896 = 119894 then
10038161003816100381610038161003816119864 119883
119905sdot (119883
119905+119894119883
119905+119894+119895119883
119905+119894+119895+119896)10038161003816100381610038161003816
=100381610038161003816100381610038161003816Cov (119883
119905 119883
119905+119894119883
119905+119894+119895119883
119905+119894+119895+119896)100381610038161003816100381610038161003816le 119862
1198944
(51)
Similarly for max119894 119895 119896 = 119896 we have
10038161003816100381610038161003816119864 (119883
119905119883
119905+119894119883
119905+119894+119895) sdot 119883
119905+119894+119895+11989610038161003816100381610038161003816
=10038161003816100381610038161003816Cov (119883
119905119883
119905+119894119883
119905+119894+119895 119883
119905+119894+119895+119896)10038161003816100381610038161003816le 119862
1198964
(52)
When max119894 119895 119896 = 119895 then
10038161003816100381610038161003816119864 (119883
119905119883
119905+119894) sdot (119883
119905+119894+119895119883
119905+119894+119895+119896)10038161003816100381610038161003816
=10038161003816100381610038161003816Cov (119883
119905119883
119905+119894 119883
119905+119894+119895119883
119905+119894+119895+119896)
+119864 (119883119905119883
119905+119894) 119864 (119883
119905+119894+119895119883
119905+119894+119895+119896)10038161003816100381610038161003816
le 1198621198954
+ 1198621198942119862
1198962
(53)
We keep on estimating the fourth moment of 119878119899by intro-
ducing 119905 isin 0 119899mdashthe index for which |119864(119883119905119883
119905+119894119883
119905+119894+119895
119883119905+119894+119895+119896
)| attains its maximum
1198641198784
119899le 4119899 sum
0le119894119895119896le119899minus1
10038161003816100381610038161003816119864 (119883
119905119883
119905+119894119883
119905+119894+119895119883
119905+119894+119895+119896)10038161003816100381610038161003816
=4119899( sum
0le119894119896le119895
[1198621198942sdot 119862
1198962+ 119862
1198954]+ sum
0le119894119895le119896
1198621198964+ sum
0le119896119895le119894
1198621198944)
= 4119899( sum
0le119894119896le119895
1198621198942sdot 119862
1198962+ 3 sum
0le119896119895le119894
1198621198944)
(54)
6 ISRN Probability and Statistics
The terms obtained in (54) may further be bounded fromabove in the following way
sum
0le119894119896le119895
1198621198942sdot 119862
1198962=
119899minus1
sum
119895=0
119895
sum
119894119896=0
1198621198942119862
1198962
le 119899
119899minus1
sum
119894119896=0
1198621198942119862
1198962= 119899(
119899minus1
sum
119894=0
1198621198942)
2
sum
0le119896119895le119894
1198621198944
=
119899minus1
sum
119894=0
119894
sum
119895119896=0
1198621198944
le
119899minus1
sum
119894=0
(119894 + 1) (119894 + 1) 1198621198944
=
119899minus1
sum
119894=0
(119894 + 1)2119862
1198944
(55)
We thus get the inequality
1198641198784
119899le 4([119899
119899minus1
sum
119894=0
1198621198942]
2
+ 3119899
119899minus1
sum
119894=0
(119894 + 1)2119862
1198944) (56)
We shall now focus on estimating 1198621199032
and 1198621199034 Since 119880
1199051
and 1198801199052
are associated uniformly [0 1] distributed rvrsquos 1198831199051
and 1198831199052
mdashas monotone functions of these rvrsquosmdashare associat-ed as well In order to bound
1198621199032
= sup0le1199051lt1199052le119899 1199052minus1199051=119903
Cov (1198831199051
1198831199052
) (57)
let us notice that from Schwarz inequality
Cov (1198831199051
1198831199052
) = 119864 (119868 [1198801199051
le 119909] 119868 [1198801199052
le 119909]) minus 1199092
le radic1199092 minus 1199092le 119909
(58)
On the other hand invoking inequalities from [13 15]
Cov (1198831199051
1198831199052
)
le sup119909isin[01]
[119875 (1198801199051
le 1199091198801199052
le 119909) minus 119875 (1198801199051
le 119909)119875 (1198801199052
le 119909)]
le 119862 sdot Cov13
(1198801199051
1198801199052
) for associated rvrsquos4 sdot Cov (119880
1199051
1198801199052
) for MTP2rvrsquos
(59)
As a consequence
1198621199032
le min 119909 119862 sdot Cov13
(1198801199051
1198801199052
) for associated rvrsquosmin 119909 4 sdot Cov (119880
1199051
1198801199052
) for MTP2rvrsquos
(60)
where 1199052minus 119905
1= 119903 Still we need the upper bound for 119862
1199034 It
turns out that119862
1199034
= sup1003816100381610038161003816100381610038161003816100381610038161003816
Cov2
prod
119894=1
(119868 [119880119905119894
le 119909] minus 119909)
4
prod
119894=3
(119868 [119880119905119894
le 119909] minus 119909)
1003816100381610038161003816100381610038161003816100381610038161003816
(61)
In other words among all divisions of the group 119883119905119894
119894 isin
1 2 3 4 into two subgroups 1198621199034
attains the biggest valuein case we take two subgroups consisted of two rvrsquos Thesupremum runs over the set 119905
1 119905
2 119905
3 119905
4isin N 0 le 119905
1lt 119905
2lt
1199053lt 119905
4le 119899 and 119905
3minus 119905
2= 119903 Elementary calculation leads to the
following formula
1198621199034
= sup 10038161003816100381610038161003816119875 (119880119905119894
le 119909 119894 isin 1 2 3 4)
minus 119875 (119880119905119894
le 119909 119894 isin 1 2) 119875 (119880119905119894
le 119909 119894 isin 3 4)
minus 119909 [119875 (119880119905119894
le 119909 119894 isin 1 2 3)
minus 119875 (119880119905119894
le 119909 119894 isin 1 2) 119875 (1198801199053
le 119909)
+ 119875 (119880119905119894
le 119909 119894 isin 1 3 4)
minus 119875 (119880119905119894
le 119909 119894 isin 3 4) 119875 (1198801199051
le 119909)
+ 119875 (119880119905119894
le 119909 119894 isin 1 2 4)
minus 119875 (119880119905119894
le 119909 119894 isin 1 2) 119875 (1198801199054
le 119909)
+ 119875 (119880119905119894
le 119909 119894 isin 2 3 4)
minus119875 (119880119905119894
le 119909 119894 isin 3 4) 119875 (1198801199052
le 119909)]
+ 1199092[119875 (119880
1199051
le 1199091198801199053
le 119909)
minus 119875 (1198801199051
le 119909)119875 (1198801199053
le 119909)
+ 119875 (1198801199051
le 1199091198801199054
le 119909)
minus 119875 (1198801199051
le 119909)119875 (1198801199054
le 119909)
+ 119875 (1198801199052
le 1199091198801199053
le 119909)
minus 119875 (1198801199052
le 119909)119875 (1198801199053
le 119909)
+ 119875 (1198801199052
le 1199091198801199054
le 119909)
minus119875 (1198801199052
le 119909)119875 (1198801199052
le 119909)]10038161003816100381610038161003816
le10038161003816100381610038161198771
1003816100381610038161003816 +10038161003816100381610038161198772
1003816100381610038161003816 +10038161003816100381610038161198773
1003816100381610038161003816
(62)
where for the sake of simplicity 1198771 119877
2 and 119877
3are respec-
tively the free coefficient the expression with 119909 and theexpression with 119909
2 Using the Lebowitz inequality (see [15]for instance) and inequalities obtained in [13 15] we arrive at10038161003816100381610038161198771
1003816100381610038161003816 =10038161003816100381610038161003816119875 (119880
119905119894
le 119909 119894 isin 1 2 3 4)
minus119875 (119880119905119894
le 119909 119894 isin 1 2) 119875 (119880119905119894
le 119909 119894 isin 3 4)10038161003816100381610038161003816
le 11986711988011990511198801199053
+ 11986711988011990511198801199054
+ 11986711988011990521198801199053
+ 11986711988011990521198801199054
ISRN Probability and Statistics 7
le
119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
)) for associated rvrsquos
4(
2
sum
119894=1
4
sum
119895=3
Cov (119880119905119894
119880119905119895
)) for MTP2rvrsquos
(63)
Analogously we get
10038161003816100381610038161198772
1003816100381610038161003816
le
2119909 sdot 119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
)) for associated rvrsquos
2119909 sdot 4(
2
sum
119894=1
4
sum
119895=3
Cov (119880119905119894
119880119905119895
)) for MTP2rvrsquos
10038161003816100381610038161198773
1003816100381610038161003816
le
1199092sdot 119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
) ) for associated rvrsquos
1199092sdot 4(
2
sum
119894=1
4
sum
119895=3
Cov (119880119905119894
119880119905119895
)) for MTP2rvrsquos
(64)
Eventually we have
1198621199034
le
119891 (119909) sdot119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
)) for associated rvrsquos
119891 (119909) sdot4(
2
sum
119894=1
4
sum
119895=3
Cov (119880119905119894
119880119905119895
)) for MTP2rvrsquos
(65)
where 1199053minus 119905
2= 119903 and 119891(119909) = (1 + 2119909 + 119909
2) for 119909 isin [0 1]
Let us now introduce the following notation
120579119903=
sup119905isinN
Cov (119880119905 119880
119905+119903) for 119903 isin 1 2
1 for 119903 = 0
(66)
and assume it decays powerly at rate 119886 in the following way
120579119903= 119874(
1
(119903 + 1)119886) for 119886 gt 0 119903 isin 0 1 (67)
Let us get back to inequality (56) we can now carry on Atfirst for associated rvrsquos
1198641198784
119899le 4[119899
119899minus1
sum
119903=0
min 119909 119862 sdot Cov13(119880
1199051
1198801199052
)]
2
+ 4 sdot 3119899
119899minus1
sum
119903=0
(119903 + 1)2119891 (119909) sdot 119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
))
le 119863 sdot ([119899
119899minus1
sum
119903=0
min 119909 (119903 + 1)minus1198863
]
2
+119899
119899minus1
sum
119903=0
(119903 + 1)2(119903 + 1)
minus1198863)
= 119863 sdot ([119899
119899
sum
119903=1
min 119909 119903minus1198863]
2
+ 119899
119899
sum
119903=1
1199032119903minus1198863
)
le 119863 sdot ([
[
119899 sum
119903lt119909minus3119886
119909 + 119899 sum
119903ge119909minus3119886
1
1199031198863]
]
2
+ 119899
119899
sum
119903=1
1199032minus1198863
)
le 1198631sdot (119899
2119909
2(119886minus3)119886+ 120585)
(68)
where119863 and1198631are constants and
120585 = 119899
119899
sum
119903=1
1199032minus1198863
=
119874(119899) 119886 gt 9
119874 (119899 ln 119899) 119886 = 9
119874 (1198994minus1198863
) 3 lt 119886 lt 9
(69)
It is worth mentioning that in the last inequality of (68) weused the estimate
int
infin
119909
1
119905119901119889119905 sim
1
119909119901minus1for 119901 gt 1 (70)
At the same time in the case of MTP2rvrsquos we get
1198641198784
119899le 119863
2sdot (119899
2119909
2(119886minus1)119886+ 120577) (71)
where1198632is constant and
120577 = 119899
119899
sum
119903=1
1199032minus119886
=
119874(119899) 119886 gt 3
119874 (119899 ln 119899) 119886 = 3
119874 (1198994minus119886
) 1 lt 119886 lt 3
(72)
Let now 120572119899(119909) = (1radic119899)sum
119899
119894=1(119868[119880
119905119894
le 119909] minus 119909) Then
120572119899(119909) minus 120572
119899(119910) =
1
radic119899
119899
sum
119894=1
(119868 [119909 lt 119880119905119894
le 119910] minus (119909 minus 119910))
for 119909 119910 isin [0 1]
(73)
8 ISRN Probability and Statistics
has the fourth moment estimatedmdashin the case of associatedrvrsquosmdashby
119864[120572119899(119909) minus 120572
119899(119910)]
4
= 119864[1
radic119899
119899
sum
119894=1
(119868 [119909 lt 119880119905119894
le 119910] minus (119909 minus 119910))]
4
=1
1198992119864[
119899
sum
119894=1
(119868 [119909 lt 119880119905119894
le 119910] minus (119909 minus 119910))]
4
le 1198631sdot (
1
119899211989921003816100381610038161003816119909 minus 119910
1003816100381610038161003816
2(119886minus3)119886
+120585
1198992)
= 1198631sdot (
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2(119886minus3)119886
+120585
1198992)
=
119874(119899minus1) 119886 gt 9
119874(ln 119899119899
) 119886 = 9
119874 (1198992minus1198863
) 3 lt 119886 lt 9
(74)
and in the case of MTP2rvrsquos by
119864[120572119899(119909) minus 120572
119899(119910)]
4
le 1198632sdot (
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2(119886minus1)119886
+120577
1198992)
=
119874(119899minus1) 119886 gt 3
119874(ln 119899119899
) 119886 = 3
119874 (1198992minus119886
) 1 lt 119886 lt 3
(75)
In light of the Shao and Yursquos criterion our process is tight forassociated rvrsquos when 119886 gt 6 and for MTP
2rvrsquos when 119886 gt 2
Let us sum up this result in the following theorem
Theorem 3 Let 120572119899(119909) = (1radic119899)sum
119899
119894=1(119868[119880
119894le 119909] minus 119909) be the
empirical process built on an associated sequence of uniformly[0 1] distributed rvrsquos 119880
119894119894ge1
Let also
120579119903= sup
119905isinNCov (119880
119905 119880
119905+119903) = 119874 ((119903 + 1)
minus119886)
where 119903 isin 0 1 119886 gt 0
(76)
Then 120572119899(119909)
119899ge1is tight for 119886 gt 6 If the rvrsquos 119880
119894119894ge1
are MTP2
then the process is tight for 119886 gt 2
Yu assumed stationarity of 119880119894119894ge1
andsum
infin
119899=111989965+]Cov(119880
0 119880
119899) lt infin for a positive constant ]
thus the rate of decay 119886 gt 7 5 Our result weakens consid-erably these assumptions especially in the case of MTP
2rvrsquos
Louhichi in [16] proposed a different tightness criterioninvolving the so-called bracketing numbers She managedto enhance Yursquos resultmdasheven more than Shao and Yu in[8]mdashsince she proved that it suffices to take 119886 gt 4 to gettightness of the empirical process based on the associated
rvrsquos Nevertheless she kept the assumption of stationarityvalid
In the final analysis our resultrsquos advantage is the absenceof the stationarity assumption and the rate of decay for120579119903remains (up to the authorrsquos knowledge) unimproved for
MTP2rvrsquos
Unfortunately with a view to obtainingweak convergenceof the process in question that is also convergence offinite-dimensional distributions we do not know how tomanage without the assumption of stationarityTherefore weconclude with the following corollary
Corollary 4 Let 120572119899(119909) = (1radic119899)sum
119899
119894=1(119868[119880
119894le 119909] minus 119909) be the
empirical process built on a stationary associated sequence ofuniformly [0 1] distributed rvrsquos 119880
119894119894ge1
Let also
120579119903= Cov (119880
1 119880
1+119903) = 119874 ((119903 + 1)
minus119886)
where 119903 isin 0 1 119886 gt 0
(77)
Then if 119886 gt 6
120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (78)
where 119861(sdot) is the zero mean Gaussian process on [0 1] withcovariance structure defined by
1205902(119909 119910) = 119909 and 119910 minus 119909119910
+
infin
sum
119895=1
[ Cov (119868 [1198801le 119909] 119868 [119880
119895+1le 119910])
+ Cov (119868 [1198801le 119910] 119868 [119880
119895+1le 119909])]
(79)
If the rvrsquos 119880119894119894ge1
are MTP2 then it suffices to be 119886 gt 2 in
order to claim the above convergence
Proof It remains to establish convergence of finite-dimen-sional distributions repeating the procedure from [7]
4 Tightness of the Kernel-TypeEmpirical Process
In this section we shall weaken assumption imposed on thecovariance structure of rvrsquos 119883
119895119895ge1
by Cai and Roussas in [1]for the kernel estimator of the df
119865119899(119909) =
1
119899
119899
sum
119895=1
119870(
119909 minus 119883119895
ℎ119899
) (80)
They deal with a stationary sequence of negatively associatedrvrsquos (cf [17]) and need the same condition as Yu [7] that is
1003816100381610038161003816Cov (1198831 119883
1+119899)1003816100381610038161003816 = 119874(
1
11989975+]) (81)
to get tightness of the smooth empirical process (see condi-tion (A4) in [1])
ISRN Probability and Statistics 9
It turns out that it suffices to have
1003816100381610038161003816Cov (1198801 119880
1+119899)1003816100381610038161003816 = 119874(
1
1198993119901(119901minus1)) (82)
where 119901 gt 4 is a positive constant taken from the tightnesscriterion (8) It is easy to see that asymptotically we get therate 3
On the way to prove it we will also take use of aRosenthal-type inequality due to Shao andYu (seeTheorem 2in [8]) we shall recall in the following lemma
Lemma 5 Let 119901 gt 2 and 119891 be a real valued function boundedby 1 with bounded first derivative Suppose that 119883
119899119899ge1
is asequence of stationary and associated rvrsquos such that for 119899 isin N
Cov (1198831 119883
119899) = 119874 (119899
minus119887) for some 119887 gt 119901 minus 1 (83)
Then for any 120583 gt 0 there exists some positive constant 119896120583
independent of the function 119891 for which
1198641003816100381610038161003816119878119899
(119891) minus 119864119878119899(119891)
1003816100381610038161003816
119901
le 119896120583(119899
1+12058310038171003817100381710038171003817119891
101584010038171003817100381710038171003817
2
+1198991199012 [
[
119899
sum
119895=1
10038161003816100381610038161003816Cov (119891 (119883
1)119891 (119883
119895))10038161003816100381610038161003816]
]
)
(84)
As we can see the lemma assumes association but itworks for negatively associated rvrsquos as well since in the proofit reaches back the result of Newman (see Proposition 15 in[18]) where both types of association are allowed
Let us recall that 120572119899(119909) = radic119899(119865
119899(119909) minus 119864119865
119899(119909)) where
119865119899(119909) = (1119899)sum
119899
119895=1119870((119909 minus 119883
119895)ℎ
119899) It is easy to see that
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
= 119899minus1199012
11986410038161003816100381610038161003816119878119899(119891 (119883
119895)) minus 119864119878
119899(119891 (119883
119895))10038161003816100381610038161003816
119901
(85)
where
119891 (119883119895) = 119870(
119909 minus 119883119895
ℎ119899
) minus 119870(
119910 minus 119883119895
ℎ119899
)
119878119899(119891) =
119899
sum
119895=1
119891 (119883119895)
(86)
With an intent to use Lemma 5 we need 119891 to be boundedfrom above by 1 (which in our case is obvious) and to havebounded 119891
1015840 thus we assume
sup119905isinR
10038161003816100381610038161003816119870
1015840(119905)
10038161003816100381610038161003816
1
ℎ119899
le119862
119870
ℎ119899
lt infin (87)
Now applying inequality (84) we have
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
le119896120583119899minus1199012
1198991+120583
1198622
119870
ℎ2
119899
+1198991199012[
[
119899
sum
119895=1
10038161003816100381610038161003816Cov (119891 (119883
1)119891 (119883
119895))10038161003816100381610038161003816]
]
1199012
(88)
Newman showed in [18] that
Cov (1198911(119883) 119891
2(119884))
= ∬R2119891
1015840
1(119909) 119891
1015840
2(119910)
times [119875 (119883 le 119909 119884 le 119910) minus 119875 (119883 le 119909) 119875 (119884 le 119910)] 119889119909 119889119910
(89)
if 1198911and 119891
2are real valued functions on R having square
integrable derivatives 1198911015840
1and 119891
1015840
2 respectively and provided
that 1198911(119883) 119891
2(119884) have finite secondmoments In light of that
equation and linearity of covariance we get10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
=
1003816100381610038161003816100381610038161003816∬
R2[119875 (119883
1lt 119909 minus ℎ
119899119904 119883
119895lt 119909 minus ℎ
119899119905)
minus119875 (1198831lt 119909 minus ℎ
119899119904 119883
119895lt 119910 minus ℎ
119899119905)]
minus [119875 (1198831lt 119909 minus ℎ
119899119904) 119875 (119883
119895lt 119909 minus ℎ
119899119905)
minus119875 (1198831lt 119909 minus ℎ
119899119904) 119875 (119883
119895lt 119910 minus ℎ
119899119905)]
minus [119875 (1198831lt 119910 minus ℎ
119899119904 119883
119895lt 119909 minus ℎ
119899119905)
minus119875 (1198831lt 119910 minus ℎ
119899119904 119883
119895lt 119910 minus ℎ
119899119905)]
minus [119875 (1198831lt 119910 minus ℎ
119899119904) 119875 (119883
119895lt 119910 minus ℎ
119899119905)
minus 119875 (1198831lt 119910 minus ℎ
119899119904)
times 119875 (119883119895lt 119909 minus ℎ
119899119905)] 119896 (119904) 119896 (119905) 119889119904 119889119905
10038161003816100381610038161003816100381610038161003816
(90)
Without the loss of generality we may and do assume that119909 gt 119910 thus10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
=
1003816100381610038161003816100381610038161003816∬
R2119875 (119910 minus ℎ
119899119904 le 119883
1lt 119909 minus ℎ
119899119904
119910 minus ℎ119899119905 le 119883
119895lt 119909 minus ℎ
119899119905)
minus 119875 (119910 minus ℎ119899119904 le 119883
1lt 119909 minus ℎ
119899119904)
times 119875 (119910 minus ℎ119899119905 le 119883
119895lt 119909 minus ℎ
119899119905) 119896 (119904) 119896 (119905) 119889119904 119889119905
1003816100381610038161003816100381610038161003816
(91)
and by triangle inequality we get10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
le ∬R2[int
119909minusℎ119899119904
119910minusℎ119899119904
int
119909minusℎ119899119905
119910minusℎ119899119905
1198911198831119883119895(119906 V) 119889119906 119889V
+int
119909minusℎ119899119904
119910minusℎ119899119904
1198911198831(119906) 119889119906int
119909minusℎ119899119905
119910minusℎ119899119905
119891119883119895(V) 119889V] 119896 (119904) 119896 (119905) 119889119904 119889119905
(92)
10 ISRN Probability and Statistics
1198911198831119883119895
(119906 V) 1198911198831
(119906) and 119891119883119895
(V) are the joint pdf of [1198831 119883
119895]
and marginal pdfrsquos of rvrsquos1198831and119883
119895 respectively We need
to further assume that 1198621stands for a common upper bound
of 1198911198831
and 1198911198831119883119895
for all 119895 isin N that is
100381710038171003817100381710038171198911198831
10038171003817100381710038171003817le 119862
1
1003817100381710038171003817100381710038171198911198831119883119895
100381710038171003817100381710038171003817le 119862
1forall119895 ge 1 (93)
Then we finally obtain
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816le 2119862
1
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
where = max 1198621 119862
2
1
(94)
On the other hand proceeding like Cai and Roussas in [1] wecan shortly get
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
le
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119909 minus 119883
1
ℎ119899
) 119870(
119909 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119909 minus 119883
1
ℎ119899
) 119870(
119910 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119910 minus 119883
1
ℎ119899
) 119870(
119909 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119910 minus 119883
1
ℎ119899
) 119870(
119910 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
le 41198622
10038161003816100381610038161003816Cov (119883
1 119883
119895)10038161003816100381610038161003816
13
(95)
where 1198622is a constant relevant to the covariance inequality
for negatively associated rvrsquos (see [13])We now arrive at the following inequality
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
le max 1198621 119862
2
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
10038161003816100381610038161003816Cov(119883
1 119883
119895)10038161003816100381610038161003816
13
(96)
Assuming |Cov(119891(1198831) 119891(119883
119895))| = 119874(119895
minus119903) and proceeding
similarly to (68) we can get
[
[
119899
sum
119895=1
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816]
]
1199012
le max 1198621 119862
2 [
[
sum
119895lt|119909minus119910|minus2(1199033)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
+ sum
119895ge|119909minus119910|minus2(1199033)
1
1198951199033]
]
1199012
le 2max 1198621 119862
21003816100381610038161003816119909 minus 119910
1003816100381610038161003816
((119903minus3)119903)119901
(97)
To sum up we obtain the following inequality
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
le 119896120583119899
1+120583minus1199012119862
2
119870
ℎ2
119899
+ 2max 1198621 119862
21003816100381610038161003816119909 minus 119910
1003816100381610038161003816
((119903minus3)119903)119901
le 1198621198991+120583minus1199012 1
ℎ2
119899
+1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
((119903minus3)119903)119901
(98)
where 119862 = 119896120583max1198622
119896 2max119862
1 119862
2 The formula of the
tightness criterion (8) implies that ((119903 minus 3)119903)119901 gt 1 so
119903 gt3119901
119901 minus 1 where 119901 gt 2 (99)
Let us recall the assumptions imposed on the bandwidhtsℎ
119899119899ge1
by Cai and Roussas in [1]B1 lim
119899rarrinfinℎ119899= 0 and ℎ
119899gt 0 for all 119899 isin N
+
B2 lim119899rarrinfin
119899ℎ119899= infin thus ℎ
119899= 119874(1119899
1minus120573) 120573 gt 0
B3 lim119899rarrinfin
119899ℎ4
119899= 0 hence ℎ
119899= 119874(1119899
14+120575) 120575 gt 0
In light of the above
1198991+120583minus1199012 1
ℎ2
119899
= 11989932+120583+2120575minus1199012
(100)
where 120583 gt 0 119901 gt 2 and 0 lt 120575 lt 34 Confrontation with thetightness criterion (8) forces
3
2+ 120583 + 2120575 minus
119901
2lt minus
1
2 (101)
which implies 119901 gt 4 Let us conclude with the followingtheorem
Theorem6 Let 120572119899(119909) = radic119899(119865
119899(119909)minus119864119865
119899(119909)) be the empirical
process built on the kernel estimator of the df119865 for a stationarysequence of negatively associated rvrsquos Assume that conditionsA1 A2 (87) B1 B2 B3 are satisfied Then provided that thetightness criterion (8) holds with 119901 gt 4 it suffices to demand
1003816100381610038161003816Cov (1198801 119880
1+119899)1003816100381610038161003816 = 119874(
1
1198993119901(119901minus1)) (102)
in order to obtain120572
119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (103)
where 119861(sdot) is the zero mean Gaussian process with covariancestructure defined by
1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)
+
infin
sum
119895=1
[ Cov (119868 [1198831le 119909] 119868 [119883
119895+1le 119910])
+ Cov (119868 [1198831le 119910] 119868 [119883
119895+1le 119909])]
(104)
and 119909 119910 isin [0 1]
Proof The remaining covergence of finite-dimensional dis-tributions of 120572
119899(119909) is established in [1]
ISRN Probability and Statistics 11
5 Weak Convergence of theRecursive Kernel-Type Empirical Processunder IID Assumption
The aim of this section is to show weak convergence of theempirical process
120572119899(119909) =
1
radic119899
119899
sum
119895=1
[119870(
119909 minus 119883119895
ℎ119895
) minus 119864119870(
119909 minus 119883119895
ℎ119895
)] (105)
built on iid rvrsquos 119883119895119895ge1
Wewill prove that120572
119899(sdot) convergesweakly to the Brownian
bridge 119861(sdot) with the following covariance structure
1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910) for 119909 119910 isin [0 1]
(106)
It turns out that the tightness criterion (8) suffices to reachthe goal Convergence of finite-dimensional distributions of120572
119899(sdot) holds and we shall show itProceeding like Cai and Roussas in [1] we need to show
that for any 119886 119887 isin R
119886120572119899(119909) + 119887120572
119899(119910) 997888rarr 119886119861 (119909) + 119887119861 (119910) in distribution
(107)
Let us introduce the notation
119884119894(119909) = 119870(
119909 minus 119883119894
ℎ119894
) minus 119864119870(119909 minus 119883
119894
ℎ119894
) (108)
and look into the covariance structure of 120572119899(sdot)
Cov (120572119899(119909) 120572
119899(119910))
= Cov( 1
radic119899
119899
sum
119894=1
119884119894(119909)
1
radic119899
119899
sum
119894=1
119884119894(119910))
=1
119899
119899
sum
119894=1
Cov (119884119894(119909) 119884
119894(119910))
=1
119899
119899
sum
119894=1
Cov(119870(119909 minus 119883
119894
ℎ119894
) 119870(119910 minus 119883
119894
ℎ119894
))
(109)
where the second equality follows from assumed indepen-dence of rvrsquos 119883
119894119894ge1
Firstly we observe that each summand converges to119865(119909and
119910) minus 119865(119909)119865(119910) since
Cov(119870(119909 minus 119883
119894
ℎ119894
) 119870(119910 minus 119883
119894
ℎ119894
))
= 119864[119870(119909 minus 119883
119894
ℎ119894
)119870(119910 minus 119883
119894
ℎ119894
)]
minus 119864119870(119909 minus 119883
119894
ℎ119894
)119864119870(119910 minus 119883
119894
ℎ119894
)
(110)
where
119864[119870(119909 minus 119883
119894
ℎ119894
)119870(119910 minus 119883
119894
ℎ119894
)]
= int
119909and119910
minusinfin
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
+ int
119909or119910
119909and119910
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
+ int
infin
119909or119910
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
(111)
and 119865 denotes the common df of the rvrsquos 119883119894119894ge1
In [6] itwas shown that
119864119870(119909 minus 119883
119894
ℎ119894
) 997888rarr 119865 (119909) (112)
and recalling that the kernel function 119870 is a df as well wearrive at the conclusion
Secondly applying Toeplitz lemma we get
Cov (120572119899(119909) 120572
119899(119910)) 997888rarr 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)
= 1205902(119909 119910) 119899 997888rarr infin
(113)
Thus
Var (119886120572119899(119909) + 119887120572
119899(119910))
= 1198862 Var (120572
119899(119909)) + 2119886119887Cov (120572
119899(119909) 120572
119899(119910))
+ 1198872 Var (120572
119899(119910))
997888rarr 1198862120590
2(119909 119909) + 2119886119887120590
2(119909 119910) + 119887
2120590
2(119910 119910) 119899 997888rarr infin
(114)
We are now on the way to prove that 119886120572119899(119909) + 119887120572
119899(119910) con-
verges in distribution to 119886119861(119909) + 119887119861(119910) sim N(0 1198862120590
2(119909 119909) +
21198861198871205902(119909 119910) + 119887
2120590
2(119910 119910)) which in other words means that
1
radic119899
infin
sum
119894=1
119886119884119894(119909) + 119887119884
119894(119910)
radicVar (119886120572119899(119909) + 119887120572
119899(119910))
997888rarr N (0 1) (115)
In order to obtain (115) we will use Lyapunov condition
lim119899rarrinfin
1
1198632+120575
119899
119899
sum
119894=1
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884119894(119910)
1003816100381610038161003816
2+120575
= 0 (116)
for 120575 = 1 where 1198632
119899= Var(sum119899
119894=1[119886119884
119894(119909) + 119887119884
119894(119910)]) Since
|119870(sdot) minus 119864119870(sdot)| le 2
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884i (119910)1003816100381610038161003816
3
le 8 (1198863+ 119887
3) (117)
Moreover 1198632
119899= 119899Var(119886120572
119899(119909) + 119887120572
119899(119910)) together with (114)
yields
1198633
119899= 119874 (119899
32) (118)
12 ISRN Probability and Statistics
Combining (117) with (118) we obtain
1
1198633
119899
119899
sum
119894=1
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884119894(119910)
1003816100381610038161003816
3
= 119874 (119899minus12
) (119)
which shows that Lyapunov condition holds and completesthe proof of convergence of finite-dimensional distributionsof 120572
119899(119909)We summarize the result of that section in the following
theorem
Theorem 7 Let 120572119899(119909) be the recursive kernel-type empirical
process defined by (105) built on iid rvrsquos 119883119895119895ge1
withmarginal df 119865 If 120572
119899(sdot) satisfies the tightness criterion (8) then
120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (120)
where 119861 is the Brownian bridge
Acknowledgment
The author would like to thank the referees for the carefulreading of the paper and helpful remarks
References
[1] Z Cai and G G Roussas ldquoWeak convergence for smoothestimator of a distribution function under negative associationrdquoStochastic Analysis and Applications vol 17 no 2 pp 145ndash1681999
[2] Z Cai andG G Roussas ldquoEfficient Estimation of a DistributionFunction under Qudrant Dependencerdquo Scandinavian Journal ofStatistics vol 25 no 1 pp 211ndash224 1998
[3] ZW Cai and G G Roussas ldquoUniform strong estimation under120572-mixing with ratesrdquo Statistics amp Probability Letters vol 15 no1 pp 47ndash55 1992
[4] G G Roussas ldquoKernel estimates under association stronguniform consistencyrdquo Statistics amp Probability Letters vol 12 no5 pp 393ndash403 1991
[5] Y F Li and S C Yang ldquoUniformly asymptotic normalityof the smooth estimation of the distribution function underassociated samplesrdquo Acta Mathematicae Applicatae Sinica vol28 no 4 pp 639ndash651 2005
[6] Y Li C Wei and S Yang ldquoThe recursive kernel distributionfunction estimator based onnegatively and positively associatedsequencesrdquo Communications in Statistics vol 39 no 20 pp3585ndash3595 2010
[7] H Yu ldquoA Glivenko-Cantelli lemma and weak convergence forempirical processes of associated sequencesrdquo ProbabilityTheoryand Related Fields vol 95 no 3 pp 357ndash370 1993
[8] Q M Shao and H Yu ldquoWeak convergence for weightedempirical processes of dependent sequencesrdquo The Annals ofProbability vol 24 no 4 pp 2098ndash2127 1996
[9] P Billingsley Convergence of Probability Measures Wiley Seriesin Probability and Statistics Probability and Statistics JohnWiley amp Sons New York NY USA 1999
[10] J D Esary F Proschan and D W Walkup ldquoAssociation ofrandom variables with applicationsrdquo Annals of MathematicalStatistics vol 38 pp 1466ndash1474 1967
[11] A Bulinski and A Shashkin Limit Theorems for AssociatedRandom Fields and Related Systems vol 10 of Advanced Serieson Statistical Science and Applied Probability World Scientific2007
[12] S Karlin and Y Rinott ldquoClasses of orderings of measures andrelated correlation inequalities I Multivariate totally positivedistributionsrdquo Journal of Multivariate Analysis vol 10 no 4 pp467ndash498 1980
[13] P Matuła and M Ziemba ldquoGeneralized covariance inequali-tiesrdquo Central European Journal of Mathematics vol 9 no 2 pp281ndash293 2011
[14] P Doukhan and S Louhichi ldquoA new weak dependence condi-tion and applications to moment inequalitiesrdquo Stochastic Proc-esses and their Applications vol 84 no 2 pp 313ndash342 1999
[15] P Matuła ldquoA note on some inequalities for certain classes ofpositively dependent random variablesrdquo Probability and Math-ematical Statistics vol 24 no 1 pp 17ndash26 2004
[16] S Louhichi ldquoWeak convergence for empirical processes of asso-ciated sequencesrdquo Annales de lrsquoInstitut Henri Poincare vol 36no 5 pp 547ndash567 2000
[17] K Joag-Dev and F Proschan ldquoNegative association of randomvariables with applicationsrdquoThe Annals of Statistics vol 11 no1 pp 286ndash295 1983
[18] C M Newman ldquoAsymptotic independence and limit theoremsfor positively and negatively dependent random variablesrdquo inInequalities in Statistics and Probability Y L Tong Ed vol 5pp 127ndash140 Institute ofMathematical StatisticsHayward CalifUSA 1984
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 ISRN Probability and Statistics
The terms obtained in (54) may further be bounded fromabove in the following way
sum
0le119894119896le119895
1198621198942sdot 119862
1198962=
119899minus1
sum
119895=0
119895
sum
119894119896=0
1198621198942119862
1198962
le 119899
119899minus1
sum
119894119896=0
1198621198942119862
1198962= 119899(
119899minus1
sum
119894=0
1198621198942)
2
sum
0le119896119895le119894
1198621198944
=
119899minus1
sum
119894=0
119894
sum
119895119896=0
1198621198944
le
119899minus1
sum
119894=0
(119894 + 1) (119894 + 1) 1198621198944
=
119899minus1
sum
119894=0
(119894 + 1)2119862
1198944
(55)
We thus get the inequality
1198641198784
119899le 4([119899
119899minus1
sum
119894=0
1198621198942]
2
+ 3119899
119899minus1
sum
119894=0
(119894 + 1)2119862
1198944) (56)
We shall now focus on estimating 1198621199032
and 1198621199034 Since 119880
1199051
and 1198801199052
are associated uniformly [0 1] distributed rvrsquos 1198831199051
and 1198831199052
mdashas monotone functions of these rvrsquosmdashare associat-ed as well In order to bound
1198621199032
= sup0le1199051lt1199052le119899 1199052minus1199051=119903
Cov (1198831199051
1198831199052
) (57)
let us notice that from Schwarz inequality
Cov (1198831199051
1198831199052
) = 119864 (119868 [1198801199051
le 119909] 119868 [1198801199052
le 119909]) minus 1199092
le radic1199092 minus 1199092le 119909
(58)
On the other hand invoking inequalities from [13 15]
Cov (1198831199051
1198831199052
)
le sup119909isin[01]
[119875 (1198801199051
le 1199091198801199052
le 119909) minus 119875 (1198801199051
le 119909)119875 (1198801199052
le 119909)]
le 119862 sdot Cov13
(1198801199051
1198801199052
) for associated rvrsquos4 sdot Cov (119880
1199051
1198801199052
) for MTP2rvrsquos
(59)
As a consequence
1198621199032
le min 119909 119862 sdot Cov13
(1198801199051
1198801199052
) for associated rvrsquosmin 119909 4 sdot Cov (119880
1199051
1198801199052
) for MTP2rvrsquos
(60)
where 1199052minus 119905
1= 119903 Still we need the upper bound for 119862
1199034 It
turns out that119862
1199034
= sup1003816100381610038161003816100381610038161003816100381610038161003816
Cov2
prod
119894=1
(119868 [119880119905119894
le 119909] minus 119909)
4
prod
119894=3
(119868 [119880119905119894
le 119909] minus 119909)
1003816100381610038161003816100381610038161003816100381610038161003816
(61)
In other words among all divisions of the group 119883119905119894
119894 isin
1 2 3 4 into two subgroups 1198621199034
attains the biggest valuein case we take two subgroups consisted of two rvrsquos Thesupremum runs over the set 119905
1 119905
2 119905
3 119905
4isin N 0 le 119905
1lt 119905
2lt
1199053lt 119905
4le 119899 and 119905
3minus 119905
2= 119903 Elementary calculation leads to the
following formula
1198621199034
= sup 10038161003816100381610038161003816119875 (119880119905119894
le 119909 119894 isin 1 2 3 4)
minus 119875 (119880119905119894
le 119909 119894 isin 1 2) 119875 (119880119905119894
le 119909 119894 isin 3 4)
minus 119909 [119875 (119880119905119894
le 119909 119894 isin 1 2 3)
minus 119875 (119880119905119894
le 119909 119894 isin 1 2) 119875 (1198801199053
le 119909)
+ 119875 (119880119905119894
le 119909 119894 isin 1 3 4)
minus 119875 (119880119905119894
le 119909 119894 isin 3 4) 119875 (1198801199051
le 119909)
+ 119875 (119880119905119894
le 119909 119894 isin 1 2 4)
minus 119875 (119880119905119894
le 119909 119894 isin 1 2) 119875 (1198801199054
le 119909)
+ 119875 (119880119905119894
le 119909 119894 isin 2 3 4)
minus119875 (119880119905119894
le 119909 119894 isin 3 4) 119875 (1198801199052
le 119909)]
+ 1199092[119875 (119880
1199051
le 1199091198801199053
le 119909)
minus 119875 (1198801199051
le 119909)119875 (1198801199053
le 119909)
+ 119875 (1198801199051
le 1199091198801199054
le 119909)
minus 119875 (1198801199051
le 119909)119875 (1198801199054
le 119909)
+ 119875 (1198801199052
le 1199091198801199053
le 119909)
minus 119875 (1198801199052
le 119909)119875 (1198801199053
le 119909)
+ 119875 (1198801199052
le 1199091198801199054
le 119909)
minus119875 (1198801199052
le 119909)119875 (1198801199052
le 119909)]10038161003816100381610038161003816
le10038161003816100381610038161198771
1003816100381610038161003816 +10038161003816100381610038161198772
1003816100381610038161003816 +10038161003816100381610038161198773
1003816100381610038161003816
(62)
where for the sake of simplicity 1198771 119877
2 and 119877
3are respec-
tively the free coefficient the expression with 119909 and theexpression with 119909
2 Using the Lebowitz inequality (see [15]for instance) and inequalities obtained in [13 15] we arrive at10038161003816100381610038161198771
1003816100381610038161003816 =10038161003816100381610038161003816119875 (119880
119905119894
le 119909 119894 isin 1 2 3 4)
minus119875 (119880119905119894
le 119909 119894 isin 1 2) 119875 (119880119905119894
le 119909 119894 isin 3 4)10038161003816100381610038161003816
le 11986711988011990511198801199053
+ 11986711988011990511198801199054
+ 11986711988011990521198801199053
+ 11986711988011990521198801199054
ISRN Probability and Statistics 7
le
119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
)) for associated rvrsquos
4(
2
sum
119894=1
4
sum
119895=3
Cov (119880119905119894
119880119905119895
)) for MTP2rvrsquos
(63)
Analogously we get
10038161003816100381610038161198772
1003816100381610038161003816
le
2119909 sdot 119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
)) for associated rvrsquos
2119909 sdot 4(
2
sum
119894=1
4
sum
119895=3
Cov (119880119905119894
119880119905119895
)) for MTP2rvrsquos
10038161003816100381610038161198773
1003816100381610038161003816
le
1199092sdot 119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
) ) for associated rvrsquos
1199092sdot 4(
2
sum
119894=1
4
sum
119895=3
Cov (119880119905119894
119880119905119895
)) for MTP2rvrsquos
(64)
Eventually we have
1198621199034
le
119891 (119909) sdot119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
)) for associated rvrsquos
119891 (119909) sdot4(
2
sum
119894=1
4
sum
119895=3
Cov (119880119905119894
119880119905119895
)) for MTP2rvrsquos
(65)
where 1199053minus 119905
2= 119903 and 119891(119909) = (1 + 2119909 + 119909
2) for 119909 isin [0 1]
Let us now introduce the following notation
120579119903=
sup119905isinN
Cov (119880119905 119880
119905+119903) for 119903 isin 1 2
1 for 119903 = 0
(66)
and assume it decays powerly at rate 119886 in the following way
120579119903= 119874(
1
(119903 + 1)119886) for 119886 gt 0 119903 isin 0 1 (67)
Let us get back to inequality (56) we can now carry on Atfirst for associated rvrsquos
1198641198784
119899le 4[119899
119899minus1
sum
119903=0
min 119909 119862 sdot Cov13(119880
1199051
1198801199052
)]
2
+ 4 sdot 3119899
119899minus1
sum
119903=0
(119903 + 1)2119891 (119909) sdot 119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
))
le 119863 sdot ([119899
119899minus1
sum
119903=0
min 119909 (119903 + 1)minus1198863
]
2
+119899
119899minus1
sum
119903=0
(119903 + 1)2(119903 + 1)
minus1198863)
= 119863 sdot ([119899
119899
sum
119903=1
min 119909 119903minus1198863]
2
+ 119899
119899
sum
119903=1
1199032119903minus1198863
)
le 119863 sdot ([
[
119899 sum
119903lt119909minus3119886
119909 + 119899 sum
119903ge119909minus3119886
1
1199031198863]
]
2
+ 119899
119899
sum
119903=1
1199032minus1198863
)
le 1198631sdot (119899
2119909
2(119886minus3)119886+ 120585)
(68)
where119863 and1198631are constants and
120585 = 119899
119899
sum
119903=1
1199032minus1198863
=
119874(119899) 119886 gt 9
119874 (119899 ln 119899) 119886 = 9
119874 (1198994minus1198863
) 3 lt 119886 lt 9
(69)
It is worth mentioning that in the last inequality of (68) weused the estimate
int
infin
119909
1
119905119901119889119905 sim
1
119909119901minus1for 119901 gt 1 (70)
At the same time in the case of MTP2rvrsquos we get
1198641198784
119899le 119863
2sdot (119899
2119909
2(119886minus1)119886+ 120577) (71)
where1198632is constant and
120577 = 119899
119899
sum
119903=1
1199032minus119886
=
119874(119899) 119886 gt 3
119874 (119899 ln 119899) 119886 = 3
119874 (1198994minus119886
) 1 lt 119886 lt 3
(72)
Let now 120572119899(119909) = (1radic119899)sum
119899
119894=1(119868[119880
119905119894
le 119909] minus 119909) Then
120572119899(119909) minus 120572
119899(119910) =
1
radic119899
119899
sum
119894=1
(119868 [119909 lt 119880119905119894
le 119910] minus (119909 minus 119910))
for 119909 119910 isin [0 1]
(73)
8 ISRN Probability and Statistics
has the fourth moment estimatedmdashin the case of associatedrvrsquosmdashby
119864[120572119899(119909) minus 120572
119899(119910)]
4
= 119864[1
radic119899
119899
sum
119894=1
(119868 [119909 lt 119880119905119894
le 119910] minus (119909 minus 119910))]
4
=1
1198992119864[
119899
sum
119894=1
(119868 [119909 lt 119880119905119894
le 119910] minus (119909 minus 119910))]
4
le 1198631sdot (
1
119899211989921003816100381610038161003816119909 minus 119910
1003816100381610038161003816
2(119886minus3)119886
+120585
1198992)
= 1198631sdot (
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2(119886minus3)119886
+120585
1198992)
=
119874(119899minus1) 119886 gt 9
119874(ln 119899119899
) 119886 = 9
119874 (1198992minus1198863
) 3 lt 119886 lt 9
(74)
and in the case of MTP2rvrsquos by
119864[120572119899(119909) minus 120572
119899(119910)]
4
le 1198632sdot (
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2(119886minus1)119886
+120577
1198992)
=
119874(119899minus1) 119886 gt 3
119874(ln 119899119899
) 119886 = 3
119874 (1198992minus119886
) 1 lt 119886 lt 3
(75)
In light of the Shao and Yursquos criterion our process is tight forassociated rvrsquos when 119886 gt 6 and for MTP
2rvrsquos when 119886 gt 2
Let us sum up this result in the following theorem
Theorem 3 Let 120572119899(119909) = (1radic119899)sum
119899
119894=1(119868[119880
119894le 119909] minus 119909) be the
empirical process built on an associated sequence of uniformly[0 1] distributed rvrsquos 119880
119894119894ge1
Let also
120579119903= sup
119905isinNCov (119880
119905 119880
119905+119903) = 119874 ((119903 + 1)
minus119886)
where 119903 isin 0 1 119886 gt 0
(76)
Then 120572119899(119909)
119899ge1is tight for 119886 gt 6 If the rvrsquos 119880
119894119894ge1
are MTP2
then the process is tight for 119886 gt 2
Yu assumed stationarity of 119880119894119894ge1
andsum
infin
119899=111989965+]Cov(119880
0 119880
119899) lt infin for a positive constant ]
thus the rate of decay 119886 gt 7 5 Our result weakens consid-erably these assumptions especially in the case of MTP
2rvrsquos
Louhichi in [16] proposed a different tightness criterioninvolving the so-called bracketing numbers She managedto enhance Yursquos resultmdasheven more than Shao and Yu in[8]mdashsince she proved that it suffices to take 119886 gt 4 to gettightness of the empirical process based on the associated
rvrsquos Nevertheless she kept the assumption of stationarityvalid
In the final analysis our resultrsquos advantage is the absenceof the stationarity assumption and the rate of decay for120579119903remains (up to the authorrsquos knowledge) unimproved for
MTP2rvrsquos
Unfortunately with a view to obtainingweak convergenceof the process in question that is also convergence offinite-dimensional distributions we do not know how tomanage without the assumption of stationarityTherefore weconclude with the following corollary
Corollary 4 Let 120572119899(119909) = (1radic119899)sum
119899
119894=1(119868[119880
119894le 119909] minus 119909) be the
empirical process built on a stationary associated sequence ofuniformly [0 1] distributed rvrsquos 119880
119894119894ge1
Let also
120579119903= Cov (119880
1 119880
1+119903) = 119874 ((119903 + 1)
minus119886)
where 119903 isin 0 1 119886 gt 0
(77)
Then if 119886 gt 6
120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (78)
where 119861(sdot) is the zero mean Gaussian process on [0 1] withcovariance structure defined by
1205902(119909 119910) = 119909 and 119910 minus 119909119910
+
infin
sum
119895=1
[ Cov (119868 [1198801le 119909] 119868 [119880
119895+1le 119910])
+ Cov (119868 [1198801le 119910] 119868 [119880
119895+1le 119909])]
(79)
If the rvrsquos 119880119894119894ge1
are MTP2 then it suffices to be 119886 gt 2 in
order to claim the above convergence
Proof It remains to establish convergence of finite-dimen-sional distributions repeating the procedure from [7]
4 Tightness of the Kernel-TypeEmpirical Process
In this section we shall weaken assumption imposed on thecovariance structure of rvrsquos 119883
119895119895ge1
by Cai and Roussas in [1]for the kernel estimator of the df
119865119899(119909) =
1
119899
119899
sum
119895=1
119870(
119909 minus 119883119895
ℎ119899
) (80)
They deal with a stationary sequence of negatively associatedrvrsquos (cf [17]) and need the same condition as Yu [7] that is
1003816100381610038161003816Cov (1198831 119883
1+119899)1003816100381610038161003816 = 119874(
1
11989975+]) (81)
to get tightness of the smooth empirical process (see condi-tion (A4) in [1])
ISRN Probability and Statistics 9
It turns out that it suffices to have
1003816100381610038161003816Cov (1198801 119880
1+119899)1003816100381610038161003816 = 119874(
1
1198993119901(119901minus1)) (82)
where 119901 gt 4 is a positive constant taken from the tightnesscriterion (8) It is easy to see that asymptotically we get therate 3
On the way to prove it we will also take use of aRosenthal-type inequality due to Shao andYu (seeTheorem 2in [8]) we shall recall in the following lemma
Lemma 5 Let 119901 gt 2 and 119891 be a real valued function boundedby 1 with bounded first derivative Suppose that 119883
119899119899ge1
is asequence of stationary and associated rvrsquos such that for 119899 isin N
Cov (1198831 119883
119899) = 119874 (119899
minus119887) for some 119887 gt 119901 minus 1 (83)
Then for any 120583 gt 0 there exists some positive constant 119896120583
independent of the function 119891 for which
1198641003816100381610038161003816119878119899
(119891) minus 119864119878119899(119891)
1003816100381610038161003816
119901
le 119896120583(119899
1+12058310038171003817100381710038171003817119891
101584010038171003817100381710038171003817
2
+1198991199012 [
[
119899
sum
119895=1
10038161003816100381610038161003816Cov (119891 (119883
1)119891 (119883
119895))10038161003816100381610038161003816]
]
)
(84)
As we can see the lemma assumes association but itworks for negatively associated rvrsquos as well since in the proofit reaches back the result of Newman (see Proposition 15 in[18]) where both types of association are allowed
Let us recall that 120572119899(119909) = radic119899(119865
119899(119909) minus 119864119865
119899(119909)) where
119865119899(119909) = (1119899)sum
119899
119895=1119870((119909 minus 119883
119895)ℎ
119899) It is easy to see that
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
= 119899minus1199012
11986410038161003816100381610038161003816119878119899(119891 (119883
119895)) minus 119864119878
119899(119891 (119883
119895))10038161003816100381610038161003816
119901
(85)
where
119891 (119883119895) = 119870(
119909 minus 119883119895
ℎ119899
) minus 119870(
119910 minus 119883119895
ℎ119899
)
119878119899(119891) =
119899
sum
119895=1
119891 (119883119895)
(86)
With an intent to use Lemma 5 we need 119891 to be boundedfrom above by 1 (which in our case is obvious) and to havebounded 119891
1015840 thus we assume
sup119905isinR
10038161003816100381610038161003816119870
1015840(119905)
10038161003816100381610038161003816
1
ℎ119899
le119862
119870
ℎ119899
lt infin (87)
Now applying inequality (84) we have
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
le119896120583119899minus1199012
1198991+120583
1198622
119870
ℎ2
119899
+1198991199012[
[
119899
sum
119895=1
10038161003816100381610038161003816Cov (119891 (119883
1)119891 (119883
119895))10038161003816100381610038161003816]
]
1199012
(88)
Newman showed in [18] that
Cov (1198911(119883) 119891
2(119884))
= ∬R2119891
1015840
1(119909) 119891
1015840
2(119910)
times [119875 (119883 le 119909 119884 le 119910) minus 119875 (119883 le 119909) 119875 (119884 le 119910)] 119889119909 119889119910
(89)
if 1198911and 119891
2are real valued functions on R having square
integrable derivatives 1198911015840
1and 119891
1015840
2 respectively and provided
that 1198911(119883) 119891
2(119884) have finite secondmoments In light of that
equation and linearity of covariance we get10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
=
1003816100381610038161003816100381610038161003816∬
R2[119875 (119883
1lt 119909 minus ℎ
119899119904 119883
119895lt 119909 minus ℎ
119899119905)
minus119875 (1198831lt 119909 minus ℎ
119899119904 119883
119895lt 119910 minus ℎ
119899119905)]
minus [119875 (1198831lt 119909 minus ℎ
119899119904) 119875 (119883
119895lt 119909 minus ℎ
119899119905)
minus119875 (1198831lt 119909 minus ℎ
119899119904) 119875 (119883
119895lt 119910 minus ℎ
119899119905)]
minus [119875 (1198831lt 119910 minus ℎ
119899119904 119883
119895lt 119909 minus ℎ
119899119905)
minus119875 (1198831lt 119910 minus ℎ
119899119904 119883
119895lt 119910 minus ℎ
119899119905)]
minus [119875 (1198831lt 119910 minus ℎ
119899119904) 119875 (119883
119895lt 119910 minus ℎ
119899119905)
minus 119875 (1198831lt 119910 minus ℎ
119899119904)
times 119875 (119883119895lt 119909 minus ℎ
119899119905)] 119896 (119904) 119896 (119905) 119889119904 119889119905
10038161003816100381610038161003816100381610038161003816
(90)
Without the loss of generality we may and do assume that119909 gt 119910 thus10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
=
1003816100381610038161003816100381610038161003816∬
R2119875 (119910 minus ℎ
119899119904 le 119883
1lt 119909 minus ℎ
119899119904
119910 minus ℎ119899119905 le 119883
119895lt 119909 minus ℎ
119899119905)
minus 119875 (119910 minus ℎ119899119904 le 119883
1lt 119909 minus ℎ
119899119904)
times 119875 (119910 minus ℎ119899119905 le 119883
119895lt 119909 minus ℎ
119899119905) 119896 (119904) 119896 (119905) 119889119904 119889119905
1003816100381610038161003816100381610038161003816
(91)
and by triangle inequality we get10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
le ∬R2[int
119909minusℎ119899119904
119910minusℎ119899119904
int
119909minusℎ119899119905
119910minusℎ119899119905
1198911198831119883119895(119906 V) 119889119906 119889V
+int
119909minusℎ119899119904
119910minusℎ119899119904
1198911198831(119906) 119889119906int
119909minusℎ119899119905
119910minusℎ119899119905
119891119883119895(V) 119889V] 119896 (119904) 119896 (119905) 119889119904 119889119905
(92)
10 ISRN Probability and Statistics
1198911198831119883119895
(119906 V) 1198911198831
(119906) and 119891119883119895
(V) are the joint pdf of [1198831 119883
119895]
and marginal pdfrsquos of rvrsquos1198831and119883
119895 respectively We need
to further assume that 1198621stands for a common upper bound
of 1198911198831
and 1198911198831119883119895
for all 119895 isin N that is
100381710038171003817100381710038171198911198831
10038171003817100381710038171003817le 119862
1
1003817100381710038171003817100381710038171198911198831119883119895
100381710038171003817100381710038171003817le 119862
1forall119895 ge 1 (93)
Then we finally obtain
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816le 2119862
1
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
where = max 1198621 119862
2
1
(94)
On the other hand proceeding like Cai and Roussas in [1] wecan shortly get
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
le
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119909 minus 119883
1
ℎ119899
) 119870(
119909 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119909 minus 119883
1
ℎ119899
) 119870(
119910 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119910 minus 119883
1
ℎ119899
) 119870(
119909 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119910 minus 119883
1
ℎ119899
) 119870(
119910 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
le 41198622
10038161003816100381610038161003816Cov (119883
1 119883
119895)10038161003816100381610038161003816
13
(95)
where 1198622is a constant relevant to the covariance inequality
for negatively associated rvrsquos (see [13])We now arrive at the following inequality
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
le max 1198621 119862
2
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
10038161003816100381610038161003816Cov(119883
1 119883
119895)10038161003816100381610038161003816
13
(96)
Assuming |Cov(119891(1198831) 119891(119883
119895))| = 119874(119895
minus119903) and proceeding
similarly to (68) we can get
[
[
119899
sum
119895=1
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816]
]
1199012
le max 1198621 119862
2 [
[
sum
119895lt|119909minus119910|minus2(1199033)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
+ sum
119895ge|119909minus119910|minus2(1199033)
1
1198951199033]
]
1199012
le 2max 1198621 119862
21003816100381610038161003816119909 minus 119910
1003816100381610038161003816
((119903minus3)119903)119901
(97)
To sum up we obtain the following inequality
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
le 119896120583119899
1+120583minus1199012119862
2
119870
ℎ2
119899
+ 2max 1198621 119862
21003816100381610038161003816119909 minus 119910
1003816100381610038161003816
((119903minus3)119903)119901
le 1198621198991+120583minus1199012 1
ℎ2
119899
+1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
((119903minus3)119903)119901
(98)
where 119862 = 119896120583max1198622
119896 2max119862
1 119862
2 The formula of the
tightness criterion (8) implies that ((119903 minus 3)119903)119901 gt 1 so
119903 gt3119901
119901 minus 1 where 119901 gt 2 (99)
Let us recall the assumptions imposed on the bandwidhtsℎ
119899119899ge1
by Cai and Roussas in [1]B1 lim
119899rarrinfinℎ119899= 0 and ℎ
119899gt 0 for all 119899 isin N
+
B2 lim119899rarrinfin
119899ℎ119899= infin thus ℎ
119899= 119874(1119899
1minus120573) 120573 gt 0
B3 lim119899rarrinfin
119899ℎ4
119899= 0 hence ℎ
119899= 119874(1119899
14+120575) 120575 gt 0
In light of the above
1198991+120583minus1199012 1
ℎ2
119899
= 11989932+120583+2120575minus1199012
(100)
where 120583 gt 0 119901 gt 2 and 0 lt 120575 lt 34 Confrontation with thetightness criterion (8) forces
3
2+ 120583 + 2120575 minus
119901
2lt minus
1
2 (101)
which implies 119901 gt 4 Let us conclude with the followingtheorem
Theorem6 Let 120572119899(119909) = radic119899(119865
119899(119909)minus119864119865
119899(119909)) be the empirical
process built on the kernel estimator of the df119865 for a stationarysequence of negatively associated rvrsquos Assume that conditionsA1 A2 (87) B1 B2 B3 are satisfied Then provided that thetightness criterion (8) holds with 119901 gt 4 it suffices to demand
1003816100381610038161003816Cov (1198801 119880
1+119899)1003816100381610038161003816 = 119874(
1
1198993119901(119901minus1)) (102)
in order to obtain120572
119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (103)
where 119861(sdot) is the zero mean Gaussian process with covariancestructure defined by
1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)
+
infin
sum
119895=1
[ Cov (119868 [1198831le 119909] 119868 [119883
119895+1le 119910])
+ Cov (119868 [1198831le 119910] 119868 [119883
119895+1le 119909])]
(104)
and 119909 119910 isin [0 1]
Proof The remaining covergence of finite-dimensional dis-tributions of 120572
119899(119909) is established in [1]
ISRN Probability and Statistics 11
5 Weak Convergence of theRecursive Kernel-Type Empirical Processunder IID Assumption
The aim of this section is to show weak convergence of theempirical process
120572119899(119909) =
1
radic119899
119899
sum
119895=1
[119870(
119909 minus 119883119895
ℎ119895
) minus 119864119870(
119909 minus 119883119895
ℎ119895
)] (105)
built on iid rvrsquos 119883119895119895ge1
Wewill prove that120572
119899(sdot) convergesweakly to the Brownian
bridge 119861(sdot) with the following covariance structure
1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910) for 119909 119910 isin [0 1]
(106)
It turns out that the tightness criterion (8) suffices to reachthe goal Convergence of finite-dimensional distributions of120572
119899(sdot) holds and we shall show itProceeding like Cai and Roussas in [1] we need to show
that for any 119886 119887 isin R
119886120572119899(119909) + 119887120572
119899(119910) 997888rarr 119886119861 (119909) + 119887119861 (119910) in distribution
(107)
Let us introduce the notation
119884119894(119909) = 119870(
119909 minus 119883119894
ℎ119894
) minus 119864119870(119909 minus 119883
119894
ℎ119894
) (108)
and look into the covariance structure of 120572119899(sdot)
Cov (120572119899(119909) 120572
119899(119910))
= Cov( 1
radic119899
119899
sum
119894=1
119884119894(119909)
1
radic119899
119899
sum
119894=1
119884119894(119910))
=1
119899
119899
sum
119894=1
Cov (119884119894(119909) 119884
119894(119910))
=1
119899
119899
sum
119894=1
Cov(119870(119909 minus 119883
119894
ℎ119894
) 119870(119910 minus 119883
119894
ℎ119894
))
(109)
where the second equality follows from assumed indepen-dence of rvrsquos 119883
119894119894ge1
Firstly we observe that each summand converges to119865(119909and
119910) minus 119865(119909)119865(119910) since
Cov(119870(119909 minus 119883
119894
ℎ119894
) 119870(119910 minus 119883
119894
ℎ119894
))
= 119864[119870(119909 minus 119883
119894
ℎ119894
)119870(119910 minus 119883
119894
ℎ119894
)]
minus 119864119870(119909 minus 119883
119894
ℎ119894
)119864119870(119910 minus 119883
119894
ℎ119894
)
(110)
where
119864[119870(119909 minus 119883
119894
ℎ119894
)119870(119910 minus 119883
119894
ℎ119894
)]
= int
119909and119910
minusinfin
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
+ int
119909or119910
119909and119910
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
+ int
infin
119909or119910
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
(111)
and 119865 denotes the common df of the rvrsquos 119883119894119894ge1
In [6] itwas shown that
119864119870(119909 minus 119883
119894
ℎ119894
) 997888rarr 119865 (119909) (112)
and recalling that the kernel function 119870 is a df as well wearrive at the conclusion
Secondly applying Toeplitz lemma we get
Cov (120572119899(119909) 120572
119899(119910)) 997888rarr 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)
= 1205902(119909 119910) 119899 997888rarr infin
(113)
Thus
Var (119886120572119899(119909) + 119887120572
119899(119910))
= 1198862 Var (120572
119899(119909)) + 2119886119887Cov (120572
119899(119909) 120572
119899(119910))
+ 1198872 Var (120572
119899(119910))
997888rarr 1198862120590
2(119909 119909) + 2119886119887120590
2(119909 119910) + 119887
2120590
2(119910 119910) 119899 997888rarr infin
(114)
We are now on the way to prove that 119886120572119899(119909) + 119887120572
119899(119910) con-
verges in distribution to 119886119861(119909) + 119887119861(119910) sim N(0 1198862120590
2(119909 119909) +
21198861198871205902(119909 119910) + 119887
2120590
2(119910 119910)) which in other words means that
1
radic119899
infin
sum
119894=1
119886119884119894(119909) + 119887119884
119894(119910)
radicVar (119886120572119899(119909) + 119887120572
119899(119910))
997888rarr N (0 1) (115)
In order to obtain (115) we will use Lyapunov condition
lim119899rarrinfin
1
1198632+120575
119899
119899
sum
119894=1
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884119894(119910)
1003816100381610038161003816
2+120575
= 0 (116)
for 120575 = 1 where 1198632
119899= Var(sum119899
119894=1[119886119884
119894(119909) + 119887119884
119894(119910)]) Since
|119870(sdot) minus 119864119870(sdot)| le 2
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884i (119910)1003816100381610038161003816
3
le 8 (1198863+ 119887
3) (117)
Moreover 1198632
119899= 119899Var(119886120572
119899(119909) + 119887120572
119899(119910)) together with (114)
yields
1198633
119899= 119874 (119899
32) (118)
12 ISRN Probability and Statistics
Combining (117) with (118) we obtain
1
1198633
119899
119899
sum
119894=1
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884119894(119910)
1003816100381610038161003816
3
= 119874 (119899minus12
) (119)
which shows that Lyapunov condition holds and completesthe proof of convergence of finite-dimensional distributionsof 120572
119899(119909)We summarize the result of that section in the following
theorem
Theorem 7 Let 120572119899(119909) be the recursive kernel-type empirical
process defined by (105) built on iid rvrsquos 119883119895119895ge1
withmarginal df 119865 If 120572
119899(sdot) satisfies the tightness criterion (8) then
120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (120)
where 119861 is the Brownian bridge
Acknowledgment
The author would like to thank the referees for the carefulreading of the paper and helpful remarks
References
[1] Z Cai and G G Roussas ldquoWeak convergence for smoothestimator of a distribution function under negative associationrdquoStochastic Analysis and Applications vol 17 no 2 pp 145ndash1681999
[2] Z Cai andG G Roussas ldquoEfficient Estimation of a DistributionFunction under Qudrant Dependencerdquo Scandinavian Journal ofStatistics vol 25 no 1 pp 211ndash224 1998
[3] ZW Cai and G G Roussas ldquoUniform strong estimation under120572-mixing with ratesrdquo Statistics amp Probability Letters vol 15 no1 pp 47ndash55 1992
[4] G G Roussas ldquoKernel estimates under association stronguniform consistencyrdquo Statistics amp Probability Letters vol 12 no5 pp 393ndash403 1991
[5] Y F Li and S C Yang ldquoUniformly asymptotic normalityof the smooth estimation of the distribution function underassociated samplesrdquo Acta Mathematicae Applicatae Sinica vol28 no 4 pp 639ndash651 2005
[6] Y Li C Wei and S Yang ldquoThe recursive kernel distributionfunction estimator based onnegatively and positively associatedsequencesrdquo Communications in Statistics vol 39 no 20 pp3585ndash3595 2010
[7] H Yu ldquoA Glivenko-Cantelli lemma and weak convergence forempirical processes of associated sequencesrdquo ProbabilityTheoryand Related Fields vol 95 no 3 pp 357ndash370 1993
[8] Q M Shao and H Yu ldquoWeak convergence for weightedempirical processes of dependent sequencesrdquo The Annals ofProbability vol 24 no 4 pp 2098ndash2127 1996
[9] P Billingsley Convergence of Probability Measures Wiley Seriesin Probability and Statistics Probability and Statistics JohnWiley amp Sons New York NY USA 1999
[10] J D Esary F Proschan and D W Walkup ldquoAssociation ofrandom variables with applicationsrdquo Annals of MathematicalStatistics vol 38 pp 1466ndash1474 1967
[11] A Bulinski and A Shashkin Limit Theorems for AssociatedRandom Fields and Related Systems vol 10 of Advanced Serieson Statistical Science and Applied Probability World Scientific2007
[12] S Karlin and Y Rinott ldquoClasses of orderings of measures andrelated correlation inequalities I Multivariate totally positivedistributionsrdquo Journal of Multivariate Analysis vol 10 no 4 pp467ndash498 1980
[13] P Matuła and M Ziemba ldquoGeneralized covariance inequali-tiesrdquo Central European Journal of Mathematics vol 9 no 2 pp281ndash293 2011
[14] P Doukhan and S Louhichi ldquoA new weak dependence condi-tion and applications to moment inequalitiesrdquo Stochastic Proc-esses and their Applications vol 84 no 2 pp 313ndash342 1999
[15] P Matuła ldquoA note on some inequalities for certain classes ofpositively dependent random variablesrdquo Probability and Math-ematical Statistics vol 24 no 1 pp 17ndash26 2004
[16] S Louhichi ldquoWeak convergence for empirical processes of asso-ciated sequencesrdquo Annales de lrsquoInstitut Henri Poincare vol 36no 5 pp 547ndash567 2000
[17] K Joag-Dev and F Proschan ldquoNegative association of randomvariables with applicationsrdquoThe Annals of Statistics vol 11 no1 pp 286ndash295 1983
[18] C M Newman ldquoAsymptotic independence and limit theoremsfor positively and negatively dependent random variablesrdquo inInequalities in Statistics and Probability Y L Tong Ed vol 5pp 127ndash140 Institute ofMathematical StatisticsHayward CalifUSA 1984
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
Volume 2014
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
ISRN Probability and Statistics 7
le
119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
)) for associated rvrsquos
4(
2
sum
119894=1
4
sum
119895=3
Cov (119880119905119894
119880119905119895
)) for MTP2rvrsquos
(63)
Analogously we get
10038161003816100381610038161198772
1003816100381610038161003816
le
2119909 sdot 119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
)) for associated rvrsquos
2119909 sdot 4(
2
sum
119894=1
4
sum
119895=3
Cov (119880119905119894
119880119905119895
)) for MTP2rvrsquos
10038161003816100381610038161198773
1003816100381610038161003816
le
1199092sdot 119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
) ) for associated rvrsquos
1199092sdot 4(
2
sum
119894=1
4
sum
119895=3
Cov (119880119905119894
119880119905119895
)) for MTP2rvrsquos
(64)
Eventually we have
1198621199034
le
119891 (119909) sdot119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
)) for associated rvrsquos
119891 (119909) sdot4(
2
sum
119894=1
4
sum
119895=3
Cov (119880119905119894
119880119905119895
)) for MTP2rvrsquos
(65)
where 1199053minus 119905
2= 119903 and 119891(119909) = (1 + 2119909 + 119909
2) for 119909 isin [0 1]
Let us now introduce the following notation
120579119903=
sup119905isinN
Cov (119880119905 119880
119905+119903) for 119903 isin 1 2
1 for 119903 = 0
(66)
and assume it decays powerly at rate 119886 in the following way
120579119903= 119874(
1
(119903 + 1)119886) for 119886 gt 0 119903 isin 0 1 (67)
Let us get back to inequality (56) we can now carry on Atfirst for associated rvrsquos
1198641198784
119899le 4[119899
119899minus1
sum
119903=0
min 119909 119862 sdot Cov13(119880
1199051
1198801199052
)]
2
+ 4 sdot 3119899
119899minus1
sum
119903=0
(119903 + 1)2119891 (119909) sdot 119862(
2
sum
119894=1
4
sum
119895=3
Cov13(119880
119905119894
119880119905119895
))
le 119863 sdot ([119899
119899minus1
sum
119903=0
min 119909 (119903 + 1)minus1198863
]
2
+119899
119899minus1
sum
119903=0
(119903 + 1)2(119903 + 1)
minus1198863)
= 119863 sdot ([119899
119899
sum
119903=1
min 119909 119903minus1198863]
2
+ 119899
119899
sum
119903=1
1199032119903minus1198863
)
le 119863 sdot ([
[
119899 sum
119903lt119909minus3119886
119909 + 119899 sum
119903ge119909minus3119886
1
1199031198863]
]
2
+ 119899
119899
sum
119903=1
1199032minus1198863
)
le 1198631sdot (119899
2119909
2(119886minus3)119886+ 120585)
(68)
where119863 and1198631are constants and
120585 = 119899
119899
sum
119903=1
1199032minus1198863
=
119874(119899) 119886 gt 9
119874 (119899 ln 119899) 119886 = 9
119874 (1198994minus1198863
) 3 lt 119886 lt 9
(69)
It is worth mentioning that in the last inequality of (68) weused the estimate
int
infin
119909
1
119905119901119889119905 sim
1
119909119901minus1for 119901 gt 1 (70)
At the same time in the case of MTP2rvrsquos we get
1198641198784
119899le 119863
2sdot (119899
2119909
2(119886minus1)119886+ 120577) (71)
where1198632is constant and
120577 = 119899
119899
sum
119903=1
1199032minus119886
=
119874(119899) 119886 gt 3
119874 (119899 ln 119899) 119886 = 3
119874 (1198994minus119886
) 1 lt 119886 lt 3
(72)
Let now 120572119899(119909) = (1radic119899)sum
119899
119894=1(119868[119880
119905119894
le 119909] minus 119909) Then
120572119899(119909) minus 120572
119899(119910) =
1
radic119899
119899
sum
119894=1
(119868 [119909 lt 119880119905119894
le 119910] minus (119909 minus 119910))
for 119909 119910 isin [0 1]
(73)
8 ISRN Probability and Statistics
has the fourth moment estimatedmdashin the case of associatedrvrsquosmdashby
119864[120572119899(119909) minus 120572
119899(119910)]
4
= 119864[1
radic119899
119899
sum
119894=1
(119868 [119909 lt 119880119905119894
le 119910] minus (119909 minus 119910))]
4
=1
1198992119864[
119899
sum
119894=1
(119868 [119909 lt 119880119905119894
le 119910] minus (119909 minus 119910))]
4
le 1198631sdot (
1
119899211989921003816100381610038161003816119909 minus 119910
1003816100381610038161003816
2(119886minus3)119886
+120585
1198992)
= 1198631sdot (
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2(119886minus3)119886
+120585
1198992)
=
119874(119899minus1) 119886 gt 9
119874(ln 119899119899
) 119886 = 9
119874 (1198992minus1198863
) 3 lt 119886 lt 9
(74)
and in the case of MTP2rvrsquos by
119864[120572119899(119909) minus 120572
119899(119910)]
4
le 1198632sdot (
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2(119886minus1)119886
+120577
1198992)
=
119874(119899minus1) 119886 gt 3
119874(ln 119899119899
) 119886 = 3
119874 (1198992minus119886
) 1 lt 119886 lt 3
(75)
In light of the Shao and Yursquos criterion our process is tight forassociated rvrsquos when 119886 gt 6 and for MTP
2rvrsquos when 119886 gt 2
Let us sum up this result in the following theorem
Theorem 3 Let 120572119899(119909) = (1radic119899)sum
119899
119894=1(119868[119880
119894le 119909] minus 119909) be the
empirical process built on an associated sequence of uniformly[0 1] distributed rvrsquos 119880
119894119894ge1
Let also
120579119903= sup
119905isinNCov (119880
119905 119880
119905+119903) = 119874 ((119903 + 1)
minus119886)
where 119903 isin 0 1 119886 gt 0
(76)
Then 120572119899(119909)
119899ge1is tight for 119886 gt 6 If the rvrsquos 119880
119894119894ge1
are MTP2
then the process is tight for 119886 gt 2
Yu assumed stationarity of 119880119894119894ge1
andsum
infin
119899=111989965+]Cov(119880
0 119880
119899) lt infin for a positive constant ]
thus the rate of decay 119886 gt 7 5 Our result weakens consid-erably these assumptions especially in the case of MTP
2rvrsquos
Louhichi in [16] proposed a different tightness criterioninvolving the so-called bracketing numbers She managedto enhance Yursquos resultmdasheven more than Shao and Yu in[8]mdashsince she proved that it suffices to take 119886 gt 4 to gettightness of the empirical process based on the associated
rvrsquos Nevertheless she kept the assumption of stationarityvalid
In the final analysis our resultrsquos advantage is the absenceof the stationarity assumption and the rate of decay for120579119903remains (up to the authorrsquos knowledge) unimproved for
MTP2rvrsquos
Unfortunately with a view to obtainingweak convergenceof the process in question that is also convergence offinite-dimensional distributions we do not know how tomanage without the assumption of stationarityTherefore weconclude with the following corollary
Corollary 4 Let 120572119899(119909) = (1radic119899)sum
119899
119894=1(119868[119880
119894le 119909] minus 119909) be the
empirical process built on a stationary associated sequence ofuniformly [0 1] distributed rvrsquos 119880
119894119894ge1
Let also
120579119903= Cov (119880
1 119880
1+119903) = 119874 ((119903 + 1)
minus119886)
where 119903 isin 0 1 119886 gt 0
(77)
Then if 119886 gt 6
120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (78)
where 119861(sdot) is the zero mean Gaussian process on [0 1] withcovariance structure defined by
1205902(119909 119910) = 119909 and 119910 minus 119909119910
+
infin
sum
119895=1
[ Cov (119868 [1198801le 119909] 119868 [119880
119895+1le 119910])
+ Cov (119868 [1198801le 119910] 119868 [119880
119895+1le 119909])]
(79)
If the rvrsquos 119880119894119894ge1
are MTP2 then it suffices to be 119886 gt 2 in
order to claim the above convergence
Proof It remains to establish convergence of finite-dimen-sional distributions repeating the procedure from [7]
4 Tightness of the Kernel-TypeEmpirical Process
In this section we shall weaken assumption imposed on thecovariance structure of rvrsquos 119883
119895119895ge1
by Cai and Roussas in [1]for the kernel estimator of the df
119865119899(119909) =
1
119899
119899
sum
119895=1
119870(
119909 minus 119883119895
ℎ119899
) (80)
They deal with a stationary sequence of negatively associatedrvrsquos (cf [17]) and need the same condition as Yu [7] that is
1003816100381610038161003816Cov (1198831 119883
1+119899)1003816100381610038161003816 = 119874(
1
11989975+]) (81)
to get tightness of the smooth empirical process (see condi-tion (A4) in [1])
ISRN Probability and Statistics 9
It turns out that it suffices to have
1003816100381610038161003816Cov (1198801 119880
1+119899)1003816100381610038161003816 = 119874(
1
1198993119901(119901minus1)) (82)
where 119901 gt 4 is a positive constant taken from the tightnesscriterion (8) It is easy to see that asymptotically we get therate 3
On the way to prove it we will also take use of aRosenthal-type inequality due to Shao andYu (seeTheorem 2in [8]) we shall recall in the following lemma
Lemma 5 Let 119901 gt 2 and 119891 be a real valued function boundedby 1 with bounded first derivative Suppose that 119883
119899119899ge1
is asequence of stationary and associated rvrsquos such that for 119899 isin N
Cov (1198831 119883
119899) = 119874 (119899
minus119887) for some 119887 gt 119901 minus 1 (83)
Then for any 120583 gt 0 there exists some positive constant 119896120583
independent of the function 119891 for which
1198641003816100381610038161003816119878119899
(119891) minus 119864119878119899(119891)
1003816100381610038161003816
119901
le 119896120583(119899
1+12058310038171003817100381710038171003817119891
101584010038171003817100381710038171003817
2
+1198991199012 [
[
119899
sum
119895=1
10038161003816100381610038161003816Cov (119891 (119883
1)119891 (119883
119895))10038161003816100381610038161003816]
]
)
(84)
As we can see the lemma assumes association but itworks for negatively associated rvrsquos as well since in the proofit reaches back the result of Newman (see Proposition 15 in[18]) where both types of association are allowed
Let us recall that 120572119899(119909) = radic119899(119865
119899(119909) minus 119864119865
119899(119909)) where
119865119899(119909) = (1119899)sum
119899
119895=1119870((119909 minus 119883
119895)ℎ
119899) It is easy to see that
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
= 119899minus1199012
11986410038161003816100381610038161003816119878119899(119891 (119883
119895)) minus 119864119878
119899(119891 (119883
119895))10038161003816100381610038161003816
119901
(85)
where
119891 (119883119895) = 119870(
119909 minus 119883119895
ℎ119899
) minus 119870(
119910 minus 119883119895
ℎ119899
)
119878119899(119891) =
119899
sum
119895=1
119891 (119883119895)
(86)
With an intent to use Lemma 5 we need 119891 to be boundedfrom above by 1 (which in our case is obvious) and to havebounded 119891
1015840 thus we assume
sup119905isinR
10038161003816100381610038161003816119870
1015840(119905)
10038161003816100381610038161003816
1
ℎ119899
le119862
119870
ℎ119899
lt infin (87)
Now applying inequality (84) we have
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
le119896120583119899minus1199012
1198991+120583
1198622
119870
ℎ2
119899
+1198991199012[
[
119899
sum
119895=1
10038161003816100381610038161003816Cov (119891 (119883
1)119891 (119883
119895))10038161003816100381610038161003816]
]
1199012
(88)
Newman showed in [18] that
Cov (1198911(119883) 119891
2(119884))
= ∬R2119891
1015840
1(119909) 119891
1015840
2(119910)
times [119875 (119883 le 119909 119884 le 119910) minus 119875 (119883 le 119909) 119875 (119884 le 119910)] 119889119909 119889119910
(89)
if 1198911and 119891
2are real valued functions on R having square
integrable derivatives 1198911015840
1and 119891
1015840
2 respectively and provided
that 1198911(119883) 119891
2(119884) have finite secondmoments In light of that
equation and linearity of covariance we get10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
=
1003816100381610038161003816100381610038161003816∬
R2[119875 (119883
1lt 119909 minus ℎ
119899119904 119883
119895lt 119909 minus ℎ
119899119905)
minus119875 (1198831lt 119909 minus ℎ
119899119904 119883
119895lt 119910 minus ℎ
119899119905)]
minus [119875 (1198831lt 119909 minus ℎ
119899119904) 119875 (119883
119895lt 119909 minus ℎ
119899119905)
minus119875 (1198831lt 119909 minus ℎ
119899119904) 119875 (119883
119895lt 119910 minus ℎ
119899119905)]
minus [119875 (1198831lt 119910 minus ℎ
119899119904 119883
119895lt 119909 minus ℎ
119899119905)
minus119875 (1198831lt 119910 minus ℎ
119899119904 119883
119895lt 119910 minus ℎ
119899119905)]
minus [119875 (1198831lt 119910 minus ℎ
119899119904) 119875 (119883
119895lt 119910 minus ℎ
119899119905)
minus 119875 (1198831lt 119910 minus ℎ
119899119904)
times 119875 (119883119895lt 119909 minus ℎ
119899119905)] 119896 (119904) 119896 (119905) 119889119904 119889119905
10038161003816100381610038161003816100381610038161003816
(90)
Without the loss of generality we may and do assume that119909 gt 119910 thus10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
=
1003816100381610038161003816100381610038161003816∬
R2119875 (119910 minus ℎ
119899119904 le 119883
1lt 119909 minus ℎ
119899119904
119910 minus ℎ119899119905 le 119883
119895lt 119909 minus ℎ
119899119905)
minus 119875 (119910 minus ℎ119899119904 le 119883
1lt 119909 minus ℎ
119899119904)
times 119875 (119910 minus ℎ119899119905 le 119883
119895lt 119909 minus ℎ
119899119905) 119896 (119904) 119896 (119905) 119889119904 119889119905
1003816100381610038161003816100381610038161003816
(91)
and by triangle inequality we get10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
le ∬R2[int
119909minusℎ119899119904
119910minusℎ119899119904
int
119909minusℎ119899119905
119910minusℎ119899119905
1198911198831119883119895(119906 V) 119889119906 119889V
+int
119909minusℎ119899119904
119910minusℎ119899119904
1198911198831(119906) 119889119906int
119909minusℎ119899119905
119910minusℎ119899119905
119891119883119895(V) 119889V] 119896 (119904) 119896 (119905) 119889119904 119889119905
(92)
10 ISRN Probability and Statistics
1198911198831119883119895
(119906 V) 1198911198831
(119906) and 119891119883119895
(V) are the joint pdf of [1198831 119883
119895]
and marginal pdfrsquos of rvrsquos1198831and119883
119895 respectively We need
to further assume that 1198621stands for a common upper bound
of 1198911198831
and 1198911198831119883119895
for all 119895 isin N that is
100381710038171003817100381710038171198911198831
10038171003817100381710038171003817le 119862
1
1003817100381710038171003817100381710038171198911198831119883119895
100381710038171003817100381710038171003817le 119862
1forall119895 ge 1 (93)
Then we finally obtain
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816le 2119862
1
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
where = max 1198621 119862
2
1
(94)
On the other hand proceeding like Cai and Roussas in [1] wecan shortly get
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
le
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119909 minus 119883
1
ℎ119899
) 119870(
119909 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119909 minus 119883
1
ℎ119899
) 119870(
119910 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119910 minus 119883
1
ℎ119899
) 119870(
119909 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119910 minus 119883
1
ℎ119899
) 119870(
119910 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
le 41198622
10038161003816100381610038161003816Cov (119883
1 119883
119895)10038161003816100381610038161003816
13
(95)
where 1198622is a constant relevant to the covariance inequality
for negatively associated rvrsquos (see [13])We now arrive at the following inequality
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
le max 1198621 119862
2
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
10038161003816100381610038161003816Cov(119883
1 119883
119895)10038161003816100381610038161003816
13
(96)
Assuming |Cov(119891(1198831) 119891(119883
119895))| = 119874(119895
minus119903) and proceeding
similarly to (68) we can get
[
[
119899
sum
119895=1
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816]
]
1199012
le max 1198621 119862
2 [
[
sum
119895lt|119909minus119910|minus2(1199033)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
+ sum
119895ge|119909minus119910|minus2(1199033)
1
1198951199033]
]
1199012
le 2max 1198621 119862
21003816100381610038161003816119909 minus 119910
1003816100381610038161003816
((119903minus3)119903)119901
(97)
To sum up we obtain the following inequality
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
le 119896120583119899
1+120583minus1199012119862
2
119870
ℎ2
119899
+ 2max 1198621 119862
21003816100381610038161003816119909 minus 119910
1003816100381610038161003816
((119903minus3)119903)119901
le 1198621198991+120583minus1199012 1
ℎ2
119899
+1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
((119903minus3)119903)119901
(98)
where 119862 = 119896120583max1198622
119896 2max119862
1 119862
2 The formula of the
tightness criterion (8) implies that ((119903 minus 3)119903)119901 gt 1 so
119903 gt3119901
119901 minus 1 where 119901 gt 2 (99)
Let us recall the assumptions imposed on the bandwidhtsℎ
119899119899ge1
by Cai and Roussas in [1]B1 lim
119899rarrinfinℎ119899= 0 and ℎ
119899gt 0 for all 119899 isin N
+
B2 lim119899rarrinfin
119899ℎ119899= infin thus ℎ
119899= 119874(1119899
1minus120573) 120573 gt 0
B3 lim119899rarrinfin
119899ℎ4
119899= 0 hence ℎ
119899= 119874(1119899
14+120575) 120575 gt 0
In light of the above
1198991+120583minus1199012 1
ℎ2
119899
= 11989932+120583+2120575minus1199012
(100)
where 120583 gt 0 119901 gt 2 and 0 lt 120575 lt 34 Confrontation with thetightness criterion (8) forces
3
2+ 120583 + 2120575 minus
119901
2lt minus
1
2 (101)
which implies 119901 gt 4 Let us conclude with the followingtheorem
Theorem6 Let 120572119899(119909) = radic119899(119865
119899(119909)minus119864119865
119899(119909)) be the empirical
process built on the kernel estimator of the df119865 for a stationarysequence of negatively associated rvrsquos Assume that conditionsA1 A2 (87) B1 B2 B3 are satisfied Then provided that thetightness criterion (8) holds with 119901 gt 4 it suffices to demand
1003816100381610038161003816Cov (1198801 119880
1+119899)1003816100381610038161003816 = 119874(
1
1198993119901(119901minus1)) (102)
in order to obtain120572
119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (103)
where 119861(sdot) is the zero mean Gaussian process with covariancestructure defined by
1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)
+
infin
sum
119895=1
[ Cov (119868 [1198831le 119909] 119868 [119883
119895+1le 119910])
+ Cov (119868 [1198831le 119910] 119868 [119883
119895+1le 119909])]
(104)
and 119909 119910 isin [0 1]
Proof The remaining covergence of finite-dimensional dis-tributions of 120572
119899(119909) is established in [1]
ISRN Probability and Statistics 11
5 Weak Convergence of theRecursive Kernel-Type Empirical Processunder IID Assumption
The aim of this section is to show weak convergence of theempirical process
120572119899(119909) =
1
radic119899
119899
sum
119895=1
[119870(
119909 minus 119883119895
ℎ119895
) minus 119864119870(
119909 minus 119883119895
ℎ119895
)] (105)
built on iid rvrsquos 119883119895119895ge1
Wewill prove that120572
119899(sdot) convergesweakly to the Brownian
bridge 119861(sdot) with the following covariance structure
1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910) for 119909 119910 isin [0 1]
(106)
It turns out that the tightness criterion (8) suffices to reachthe goal Convergence of finite-dimensional distributions of120572
119899(sdot) holds and we shall show itProceeding like Cai and Roussas in [1] we need to show
that for any 119886 119887 isin R
119886120572119899(119909) + 119887120572
119899(119910) 997888rarr 119886119861 (119909) + 119887119861 (119910) in distribution
(107)
Let us introduce the notation
119884119894(119909) = 119870(
119909 minus 119883119894
ℎ119894
) minus 119864119870(119909 minus 119883
119894
ℎ119894
) (108)
and look into the covariance structure of 120572119899(sdot)
Cov (120572119899(119909) 120572
119899(119910))
= Cov( 1
radic119899
119899
sum
119894=1
119884119894(119909)
1
radic119899
119899
sum
119894=1
119884119894(119910))
=1
119899
119899
sum
119894=1
Cov (119884119894(119909) 119884
119894(119910))
=1
119899
119899
sum
119894=1
Cov(119870(119909 minus 119883
119894
ℎ119894
) 119870(119910 minus 119883
119894
ℎ119894
))
(109)
where the second equality follows from assumed indepen-dence of rvrsquos 119883
119894119894ge1
Firstly we observe that each summand converges to119865(119909and
119910) minus 119865(119909)119865(119910) since
Cov(119870(119909 minus 119883
119894
ℎ119894
) 119870(119910 minus 119883
119894
ℎ119894
))
= 119864[119870(119909 minus 119883
119894
ℎ119894
)119870(119910 minus 119883
119894
ℎ119894
)]
minus 119864119870(119909 minus 119883
119894
ℎ119894
)119864119870(119910 minus 119883
119894
ℎ119894
)
(110)
where
119864[119870(119909 minus 119883
119894
ℎ119894
)119870(119910 minus 119883
119894
ℎ119894
)]
= int
119909and119910
minusinfin
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
+ int
119909or119910
119909and119910
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
+ int
infin
119909or119910
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
(111)
and 119865 denotes the common df of the rvrsquos 119883119894119894ge1
In [6] itwas shown that
119864119870(119909 minus 119883
119894
ℎ119894
) 997888rarr 119865 (119909) (112)
and recalling that the kernel function 119870 is a df as well wearrive at the conclusion
Secondly applying Toeplitz lemma we get
Cov (120572119899(119909) 120572
119899(119910)) 997888rarr 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)
= 1205902(119909 119910) 119899 997888rarr infin
(113)
Thus
Var (119886120572119899(119909) + 119887120572
119899(119910))
= 1198862 Var (120572
119899(119909)) + 2119886119887Cov (120572
119899(119909) 120572
119899(119910))
+ 1198872 Var (120572
119899(119910))
997888rarr 1198862120590
2(119909 119909) + 2119886119887120590
2(119909 119910) + 119887
2120590
2(119910 119910) 119899 997888rarr infin
(114)
We are now on the way to prove that 119886120572119899(119909) + 119887120572
119899(119910) con-
verges in distribution to 119886119861(119909) + 119887119861(119910) sim N(0 1198862120590
2(119909 119909) +
21198861198871205902(119909 119910) + 119887
2120590
2(119910 119910)) which in other words means that
1
radic119899
infin
sum
119894=1
119886119884119894(119909) + 119887119884
119894(119910)
radicVar (119886120572119899(119909) + 119887120572
119899(119910))
997888rarr N (0 1) (115)
In order to obtain (115) we will use Lyapunov condition
lim119899rarrinfin
1
1198632+120575
119899
119899
sum
119894=1
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884119894(119910)
1003816100381610038161003816
2+120575
= 0 (116)
for 120575 = 1 where 1198632
119899= Var(sum119899
119894=1[119886119884
119894(119909) + 119887119884
119894(119910)]) Since
|119870(sdot) minus 119864119870(sdot)| le 2
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884i (119910)1003816100381610038161003816
3
le 8 (1198863+ 119887
3) (117)
Moreover 1198632
119899= 119899Var(119886120572
119899(119909) + 119887120572
119899(119910)) together with (114)
yields
1198633
119899= 119874 (119899
32) (118)
12 ISRN Probability and Statistics
Combining (117) with (118) we obtain
1
1198633
119899
119899
sum
119894=1
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884119894(119910)
1003816100381610038161003816
3
= 119874 (119899minus12
) (119)
which shows that Lyapunov condition holds and completesthe proof of convergence of finite-dimensional distributionsof 120572
119899(119909)We summarize the result of that section in the following
theorem
Theorem 7 Let 120572119899(119909) be the recursive kernel-type empirical
process defined by (105) built on iid rvrsquos 119883119895119895ge1
withmarginal df 119865 If 120572
119899(sdot) satisfies the tightness criterion (8) then
120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (120)
where 119861 is the Brownian bridge
Acknowledgment
The author would like to thank the referees for the carefulreading of the paper and helpful remarks
References
[1] Z Cai and G G Roussas ldquoWeak convergence for smoothestimator of a distribution function under negative associationrdquoStochastic Analysis and Applications vol 17 no 2 pp 145ndash1681999
[2] Z Cai andG G Roussas ldquoEfficient Estimation of a DistributionFunction under Qudrant Dependencerdquo Scandinavian Journal ofStatistics vol 25 no 1 pp 211ndash224 1998
[3] ZW Cai and G G Roussas ldquoUniform strong estimation under120572-mixing with ratesrdquo Statistics amp Probability Letters vol 15 no1 pp 47ndash55 1992
[4] G G Roussas ldquoKernel estimates under association stronguniform consistencyrdquo Statistics amp Probability Letters vol 12 no5 pp 393ndash403 1991
[5] Y F Li and S C Yang ldquoUniformly asymptotic normalityof the smooth estimation of the distribution function underassociated samplesrdquo Acta Mathematicae Applicatae Sinica vol28 no 4 pp 639ndash651 2005
[6] Y Li C Wei and S Yang ldquoThe recursive kernel distributionfunction estimator based onnegatively and positively associatedsequencesrdquo Communications in Statistics vol 39 no 20 pp3585ndash3595 2010
[7] H Yu ldquoA Glivenko-Cantelli lemma and weak convergence forempirical processes of associated sequencesrdquo ProbabilityTheoryand Related Fields vol 95 no 3 pp 357ndash370 1993
[8] Q M Shao and H Yu ldquoWeak convergence for weightedempirical processes of dependent sequencesrdquo The Annals ofProbability vol 24 no 4 pp 2098ndash2127 1996
[9] P Billingsley Convergence of Probability Measures Wiley Seriesin Probability and Statistics Probability and Statistics JohnWiley amp Sons New York NY USA 1999
[10] J D Esary F Proschan and D W Walkup ldquoAssociation ofrandom variables with applicationsrdquo Annals of MathematicalStatistics vol 38 pp 1466ndash1474 1967
[11] A Bulinski and A Shashkin Limit Theorems for AssociatedRandom Fields and Related Systems vol 10 of Advanced Serieson Statistical Science and Applied Probability World Scientific2007
[12] S Karlin and Y Rinott ldquoClasses of orderings of measures andrelated correlation inequalities I Multivariate totally positivedistributionsrdquo Journal of Multivariate Analysis vol 10 no 4 pp467ndash498 1980
[13] P Matuła and M Ziemba ldquoGeneralized covariance inequali-tiesrdquo Central European Journal of Mathematics vol 9 no 2 pp281ndash293 2011
[14] P Doukhan and S Louhichi ldquoA new weak dependence condi-tion and applications to moment inequalitiesrdquo Stochastic Proc-esses and their Applications vol 84 no 2 pp 313ndash342 1999
[15] P Matuła ldquoA note on some inequalities for certain classes ofpositively dependent random variablesrdquo Probability and Math-ematical Statistics vol 24 no 1 pp 17ndash26 2004
[16] S Louhichi ldquoWeak convergence for empirical processes of asso-ciated sequencesrdquo Annales de lrsquoInstitut Henri Poincare vol 36no 5 pp 547ndash567 2000
[17] K Joag-Dev and F Proschan ldquoNegative association of randomvariables with applicationsrdquoThe Annals of Statistics vol 11 no1 pp 286ndash295 1983
[18] C M Newman ldquoAsymptotic independence and limit theoremsfor positively and negatively dependent random variablesrdquo inInequalities in Statistics and Probability Y L Tong Ed vol 5pp 127ndash140 Institute ofMathematical StatisticsHayward CalifUSA 1984
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
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8 ISRN Probability and Statistics
has the fourth moment estimatedmdashin the case of associatedrvrsquosmdashby
119864[120572119899(119909) minus 120572
119899(119910)]
4
= 119864[1
radic119899
119899
sum
119894=1
(119868 [119909 lt 119880119905119894
le 119910] minus (119909 minus 119910))]
4
=1
1198992119864[
119899
sum
119894=1
(119868 [119909 lt 119880119905119894
le 119910] minus (119909 minus 119910))]
4
le 1198631sdot (
1
119899211989921003816100381610038161003816119909 minus 119910
1003816100381610038161003816
2(119886minus3)119886
+120585
1198992)
= 1198631sdot (
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2(119886minus3)119886
+120585
1198992)
=
119874(119899minus1) 119886 gt 9
119874(ln 119899119899
) 119886 = 9
119874 (1198992minus1198863
) 3 lt 119886 lt 9
(74)
and in the case of MTP2rvrsquos by
119864[120572119899(119909) minus 120572
119899(119910)]
4
le 1198632sdot (
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2(119886minus1)119886
+120577
1198992)
=
119874(119899minus1) 119886 gt 3
119874(ln 119899119899
) 119886 = 3
119874 (1198992minus119886
) 1 lt 119886 lt 3
(75)
In light of the Shao and Yursquos criterion our process is tight forassociated rvrsquos when 119886 gt 6 and for MTP
2rvrsquos when 119886 gt 2
Let us sum up this result in the following theorem
Theorem 3 Let 120572119899(119909) = (1radic119899)sum
119899
119894=1(119868[119880
119894le 119909] minus 119909) be the
empirical process built on an associated sequence of uniformly[0 1] distributed rvrsquos 119880
119894119894ge1
Let also
120579119903= sup
119905isinNCov (119880
119905 119880
119905+119903) = 119874 ((119903 + 1)
minus119886)
where 119903 isin 0 1 119886 gt 0
(76)
Then 120572119899(119909)
119899ge1is tight for 119886 gt 6 If the rvrsquos 119880
119894119894ge1
are MTP2
then the process is tight for 119886 gt 2
Yu assumed stationarity of 119880119894119894ge1
andsum
infin
119899=111989965+]Cov(119880
0 119880
119899) lt infin for a positive constant ]
thus the rate of decay 119886 gt 7 5 Our result weakens consid-erably these assumptions especially in the case of MTP
2rvrsquos
Louhichi in [16] proposed a different tightness criterioninvolving the so-called bracketing numbers She managedto enhance Yursquos resultmdasheven more than Shao and Yu in[8]mdashsince she proved that it suffices to take 119886 gt 4 to gettightness of the empirical process based on the associated
rvrsquos Nevertheless she kept the assumption of stationarityvalid
In the final analysis our resultrsquos advantage is the absenceof the stationarity assumption and the rate of decay for120579119903remains (up to the authorrsquos knowledge) unimproved for
MTP2rvrsquos
Unfortunately with a view to obtainingweak convergenceof the process in question that is also convergence offinite-dimensional distributions we do not know how tomanage without the assumption of stationarityTherefore weconclude with the following corollary
Corollary 4 Let 120572119899(119909) = (1radic119899)sum
119899
119894=1(119868[119880
119894le 119909] minus 119909) be the
empirical process built on a stationary associated sequence ofuniformly [0 1] distributed rvrsquos 119880
119894119894ge1
Let also
120579119903= Cov (119880
1 119880
1+119903) = 119874 ((119903 + 1)
minus119886)
where 119903 isin 0 1 119886 gt 0
(77)
Then if 119886 gt 6
120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (78)
where 119861(sdot) is the zero mean Gaussian process on [0 1] withcovariance structure defined by
1205902(119909 119910) = 119909 and 119910 minus 119909119910
+
infin
sum
119895=1
[ Cov (119868 [1198801le 119909] 119868 [119880
119895+1le 119910])
+ Cov (119868 [1198801le 119910] 119868 [119880
119895+1le 119909])]
(79)
If the rvrsquos 119880119894119894ge1
are MTP2 then it suffices to be 119886 gt 2 in
order to claim the above convergence
Proof It remains to establish convergence of finite-dimen-sional distributions repeating the procedure from [7]
4 Tightness of the Kernel-TypeEmpirical Process
In this section we shall weaken assumption imposed on thecovariance structure of rvrsquos 119883
119895119895ge1
by Cai and Roussas in [1]for the kernel estimator of the df
119865119899(119909) =
1
119899
119899
sum
119895=1
119870(
119909 minus 119883119895
ℎ119899
) (80)
They deal with a stationary sequence of negatively associatedrvrsquos (cf [17]) and need the same condition as Yu [7] that is
1003816100381610038161003816Cov (1198831 119883
1+119899)1003816100381610038161003816 = 119874(
1
11989975+]) (81)
to get tightness of the smooth empirical process (see condi-tion (A4) in [1])
ISRN Probability and Statistics 9
It turns out that it suffices to have
1003816100381610038161003816Cov (1198801 119880
1+119899)1003816100381610038161003816 = 119874(
1
1198993119901(119901minus1)) (82)
where 119901 gt 4 is a positive constant taken from the tightnesscriterion (8) It is easy to see that asymptotically we get therate 3
On the way to prove it we will also take use of aRosenthal-type inequality due to Shao andYu (seeTheorem 2in [8]) we shall recall in the following lemma
Lemma 5 Let 119901 gt 2 and 119891 be a real valued function boundedby 1 with bounded first derivative Suppose that 119883
119899119899ge1
is asequence of stationary and associated rvrsquos such that for 119899 isin N
Cov (1198831 119883
119899) = 119874 (119899
minus119887) for some 119887 gt 119901 minus 1 (83)
Then for any 120583 gt 0 there exists some positive constant 119896120583
independent of the function 119891 for which
1198641003816100381610038161003816119878119899
(119891) minus 119864119878119899(119891)
1003816100381610038161003816
119901
le 119896120583(119899
1+12058310038171003817100381710038171003817119891
101584010038171003817100381710038171003817
2
+1198991199012 [
[
119899
sum
119895=1
10038161003816100381610038161003816Cov (119891 (119883
1)119891 (119883
119895))10038161003816100381610038161003816]
]
)
(84)
As we can see the lemma assumes association but itworks for negatively associated rvrsquos as well since in the proofit reaches back the result of Newman (see Proposition 15 in[18]) where both types of association are allowed
Let us recall that 120572119899(119909) = radic119899(119865
119899(119909) minus 119864119865
119899(119909)) where
119865119899(119909) = (1119899)sum
119899
119895=1119870((119909 minus 119883
119895)ℎ
119899) It is easy to see that
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
= 119899minus1199012
11986410038161003816100381610038161003816119878119899(119891 (119883
119895)) minus 119864119878
119899(119891 (119883
119895))10038161003816100381610038161003816
119901
(85)
where
119891 (119883119895) = 119870(
119909 minus 119883119895
ℎ119899
) minus 119870(
119910 minus 119883119895
ℎ119899
)
119878119899(119891) =
119899
sum
119895=1
119891 (119883119895)
(86)
With an intent to use Lemma 5 we need 119891 to be boundedfrom above by 1 (which in our case is obvious) and to havebounded 119891
1015840 thus we assume
sup119905isinR
10038161003816100381610038161003816119870
1015840(119905)
10038161003816100381610038161003816
1
ℎ119899
le119862
119870
ℎ119899
lt infin (87)
Now applying inequality (84) we have
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
le119896120583119899minus1199012
1198991+120583
1198622
119870
ℎ2
119899
+1198991199012[
[
119899
sum
119895=1
10038161003816100381610038161003816Cov (119891 (119883
1)119891 (119883
119895))10038161003816100381610038161003816]
]
1199012
(88)
Newman showed in [18] that
Cov (1198911(119883) 119891
2(119884))
= ∬R2119891
1015840
1(119909) 119891
1015840
2(119910)
times [119875 (119883 le 119909 119884 le 119910) minus 119875 (119883 le 119909) 119875 (119884 le 119910)] 119889119909 119889119910
(89)
if 1198911and 119891
2are real valued functions on R having square
integrable derivatives 1198911015840
1and 119891
1015840
2 respectively and provided
that 1198911(119883) 119891
2(119884) have finite secondmoments In light of that
equation and linearity of covariance we get10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
=
1003816100381610038161003816100381610038161003816∬
R2[119875 (119883
1lt 119909 minus ℎ
119899119904 119883
119895lt 119909 minus ℎ
119899119905)
minus119875 (1198831lt 119909 minus ℎ
119899119904 119883
119895lt 119910 minus ℎ
119899119905)]
minus [119875 (1198831lt 119909 minus ℎ
119899119904) 119875 (119883
119895lt 119909 minus ℎ
119899119905)
minus119875 (1198831lt 119909 minus ℎ
119899119904) 119875 (119883
119895lt 119910 minus ℎ
119899119905)]
minus [119875 (1198831lt 119910 minus ℎ
119899119904 119883
119895lt 119909 minus ℎ
119899119905)
minus119875 (1198831lt 119910 minus ℎ
119899119904 119883
119895lt 119910 minus ℎ
119899119905)]
minus [119875 (1198831lt 119910 minus ℎ
119899119904) 119875 (119883
119895lt 119910 minus ℎ
119899119905)
minus 119875 (1198831lt 119910 minus ℎ
119899119904)
times 119875 (119883119895lt 119909 minus ℎ
119899119905)] 119896 (119904) 119896 (119905) 119889119904 119889119905
10038161003816100381610038161003816100381610038161003816
(90)
Without the loss of generality we may and do assume that119909 gt 119910 thus10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
=
1003816100381610038161003816100381610038161003816∬
R2119875 (119910 minus ℎ
119899119904 le 119883
1lt 119909 minus ℎ
119899119904
119910 minus ℎ119899119905 le 119883
119895lt 119909 minus ℎ
119899119905)
minus 119875 (119910 minus ℎ119899119904 le 119883
1lt 119909 minus ℎ
119899119904)
times 119875 (119910 minus ℎ119899119905 le 119883
119895lt 119909 minus ℎ
119899119905) 119896 (119904) 119896 (119905) 119889119904 119889119905
1003816100381610038161003816100381610038161003816
(91)
and by triangle inequality we get10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
le ∬R2[int
119909minusℎ119899119904
119910minusℎ119899119904
int
119909minusℎ119899119905
119910minusℎ119899119905
1198911198831119883119895(119906 V) 119889119906 119889V
+int
119909minusℎ119899119904
119910minusℎ119899119904
1198911198831(119906) 119889119906int
119909minusℎ119899119905
119910minusℎ119899119905
119891119883119895(V) 119889V] 119896 (119904) 119896 (119905) 119889119904 119889119905
(92)
10 ISRN Probability and Statistics
1198911198831119883119895
(119906 V) 1198911198831
(119906) and 119891119883119895
(V) are the joint pdf of [1198831 119883
119895]
and marginal pdfrsquos of rvrsquos1198831and119883
119895 respectively We need
to further assume that 1198621stands for a common upper bound
of 1198911198831
and 1198911198831119883119895
for all 119895 isin N that is
100381710038171003817100381710038171198911198831
10038171003817100381710038171003817le 119862
1
1003817100381710038171003817100381710038171198911198831119883119895
100381710038171003817100381710038171003817le 119862
1forall119895 ge 1 (93)
Then we finally obtain
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816le 2119862
1
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
where = max 1198621 119862
2
1
(94)
On the other hand proceeding like Cai and Roussas in [1] wecan shortly get
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
le
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119909 minus 119883
1
ℎ119899
) 119870(
119909 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119909 minus 119883
1
ℎ119899
) 119870(
119910 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119910 minus 119883
1
ℎ119899
) 119870(
119909 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119910 minus 119883
1
ℎ119899
) 119870(
119910 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
le 41198622
10038161003816100381610038161003816Cov (119883
1 119883
119895)10038161003816100381610038161003816
13
(95)
where 1198622is a constant relevant to the covariance inequality
for negatively associated rvrsquos (see [13])We now arrive at the following inequality
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
le max 1198621 119862
2
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
10038161003816100381610038161003816Cov(119883
1 119883
119895)10038161003816100381610038161003816
13
(96)
Assuming |Cov(119891(1198831) 119891(119883
119895))| = 119874(119895
minus119903) and proceeding
similarly to (68) we can get
[
[
119899
sum
119895=1
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816]
]
1199012
le max 1198621 119862
2 [
[
sum
119895lt|119909minus119910|minus2(1199033)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
+ sum
119895ge|119909minus119910|minus2(1199033)
1
1198951199033]
]
1199012
le 2max 1198621 119862
21003816100381610038161003816119909 minus 119910
1003816100381610038161003816
((119903minus3)119903)119901
(97)
To sum up we obtain the following inequality
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
le 119896120583119899
1+120583minus1199012119862
2
119870
ℎ2
119899
+ 2max 1198621 119862
21003816100381610038161003816119909 minus 119910
1003816100381610038161003816
((119903minus3)119903)119901
le 1198621198991+120583minus1199012 1
ℎ2
119899
+1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
((119903minus3)119903)119901
(98)
where 119862 = 119896120583max1198622
119896 2max119862
1 119862
2 The formula of the
tightness criterion (8) implies that ((119903 minus 3)119903)119901 gt 1 so
119903 gt3119901
119901 minus 1 where 119901 gt 2 (99)
Let us recall the assumptions imposed on the bandwidhtsℎ
119899119899ge1
by Cai and Roussas in [1]B1 lim
119899rarrinfinℎ119899= 0 and ℎ
119899gt 0 for all 119899 isin N
+
B2 lim119899rarrinfin
119899ℎ119899= infin thus ℎ
119899= 119874(1119899
1minus120573) 120573 gt 0
B3 lim119899rarrinfin
119899ℎ4
119899= 0 hence ℎ
119899= 119874(1119899
14+120575) 120575 gt 0
In light of the above
1198991+120583minus1199012 1
ℎ2
119899
= 11989932+120583+2120575minus1199012
(100)
where 120583 gt 0 119901 gt 2 and 0 lt 120575 lt 34 Confrontation with thetightness criterion (8) forces
3
2+ 120583 + 2120575 minus
119901
2lt minus
1
2 (101)
which implies 119901 gt 4 Let us conclude with the followingtheorem
Theorem6 Let 120572119899(119909) = radic119899(119865
119899(119909)minus119864119865
119899(119909)) be the empirical
process built on the kernel estimator of the df119865 for a stationarysequence of negatively associated rvrsquos Assume that conditionsA1 A2 (87) B1 B2 B3 are satisfied Then provided that thetightness criterion (8) holds with 119901 gt 4 it suffices to demand
1003816100381610038161003816Cov (1198801 119880
1+119899)1003816100381610038161003816 = 119874(
1
1198993119901(119901minus1)) (102)
in order to obtain120572
119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (103)
where 119861(sdot) is the zero mean Gaussian process with covariancestructure defined by
1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)
+
infin
sum
119895=1
[ Cov (119868 [1198831le 119909] 119868 [119883
119895+1le 119910])
+ Cov (119868 [1198831le 119910] 119868 [119883
119895+1le 119909])]
(104)
and 119909 119910 isin [0 1]
Proof The remaining covergence of finite-dimensional dis-tributions of 120572
119899(119909) is established in [1]
ISRN Probability and Statistics 11
5 Weak Convergence of theRecursive Kernel-Type Empirical Processunder IID Assumption
The aim of this section is to show weak convergence of theempirical process
120572119899(119909) =
1
radic119899
119899
sum
119895=1
[119870(
119909 minus 119883119895
ℎ119895
) minus 119864119870(
119909 minus 119883119895
ℎ119895
)] (105)
built on iid rvrsquos 119883119895119895ge1
Wewill prove that120572
119899(sdot) convergesweakly to the Brownian
bridge 119861(sdot) with the following covariance structure
1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910) for 119909 119910 isin [0 1]
(106)
It turns out that the tightness criterion (8) suffices to reachthe goal Convergence of finite-dimensional distributions of120572
119899(sdot) holds and we shall show itProceeding like Cai and Roussas in [1] we need to show
that for any 119886 119887 isin R
119886120572119899(119909) + 119887120572
119899(119910) 997888rarr 119886119861 (119909) + 119887119861 (119910) in distribution
(107)
Let us introduce the notation
119884119894(119909) = 119870(
119909 minus 119883119894
ℎ119894
) minus 119864119870(119909 minus 119883
119894
ℎ119894
) (108)
and look into the covariance structure of 120572119899(sdot)
Cov (120572119899(119909) 120572
119899(119910))
= Cov( 1
radic119899
119899
sum
119894=1
119884119894(119909)
1
radic119899
119899
sum
119894=1
119884119894(119910))
=1
119899
119899
sum
119894=1
Cov (119884119894(119909) 119884
119894(119910))
=1
119899
119899
sum
119894=1
Cov(119870(119909 minus 119883
119894
ℎ119894
) 119870(119910 minus 119883
119894
ℎ119894
))
(109)
where the second equality follows from assumed indepen-dence of rvrsquos 119883
119894119894ge1
Firstly we observe that each summand converges to119865(119909and
119910) minus 119865(119909)119865(119910) since
Cov(119870(119909 minus 119883
119894
ℎ119894
) 119870(119910 minus 119883
119894
ℎ119894
))
= 119864[119870(119909 minus 119883
119894
ℎ119894
)119870(119910 minus 119883
119894
ℎ119894
)]
minus 119864119870(119909 minus 119883
119894
ℎ119894
)119864119870(119910 minus 119883
119894
ℎ119894
)
(110)
where
119864[119870(119909 minus 119883
119894
ℎ119894
)119870(119910 minus 119883
119894
ℎ119894
)]
= int
119909and119910
minusinfin
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
+ int
119909or119910
119909and119910
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
+ int
infin
119909or119910
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
(111)
and 119865 denotes the common df of the rvrsquos 119883119894119894ge1
In [6] itwas shown that
119864119870(119909 minus 119883
119894
ℎ119894
) 997888rarr 119865 (119909) (112)
and recalling that the kernel function 119870 is a df as well wearrive at the conclusion
Secondly applying Toeplitz lemma we get
Cov (120572119899(119909) 120572
119899(119910)) 997888rarr 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)
= 1205902(119909 119910) 119899 997888rarr infin
(113)
Thus
Var (119886120572119899(119909) + 119887120572
119899(119910))
= 1198862 Var (120572
119899(119909)) + 2119886119887Cov (120572
119899(119909) 120572
119899(119910))
+ 1198872 Var (120572
119899(119910))
997888rarr 1198862120590
2(119909 119909) + 2119886119887120590
2(119909 119910) + 119887
2120590
2(119910 119910) 119899 997888rarr infin
(114)
We are now on the way to prove that 119886120572119899(119909) + 119887120572
119899(119910) con-
verges in distribution to 119886119861(119909) + 119887119861(119910) sim N(0 1198862120590
2(119909 119909) +
21198861198871205902(119909 119910) + 119887
2120590
2(119910 119910)) which in other words means that
1
radic119899
infin
sum
119894=1
119886119884119894(119909) + 119887119884
119894(119910)
radicVar (119886120572119899(119909) + 119887120572
119899(119910))
997888rarr N (0 1) (115)
In order to obtain (115) we will use Lyapunov condition
lim119899rarrinfin
1
1198632+120575
119899
119899
sum
119894=1
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884119894(119910)
1003816100381610038161003816
2+120575
= 0 (116)
for 120575 = 1 where 1198632
119899= Var(sum119899
119894=1[119886119884
119894(119909) + 119887119884
119894(119910)]) Since
|119870(sdot) minus 119864119870(sdot)| le 2
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884i (119910)1003816100381610038161003816
3
le 8 (1198863+ 119887
3) (117)
Moreover 1198632
119899= 119899Var(119886120572
119899(119909) + 119887120572
119899(119910)) together with (114)
yields
1198633
119899= 119874 (119899
32) (118)
12 ISRN Probability and Statistics
Combining (117) with (118) we obtain
1
1198633
119899
119899
sum
119894=1
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884119894(119910)
1003816100381610038161003816
3
= 119874 (119899minus12
) (119)
which shows that Lyapunov condition holds and completesthe proof of convergence of finite-dimensional distributionsof 120572
119899(119909)We summarize the result of that section in the following
theorem
Theorem 7 Let 120572119899(119909) be the recursive kernel-type empirical
process defined by (105) built on iid rvrsquos 119883119895119895ge1
withmarginal df 119865 If 120572
119899(sdot) satisfies the tightness criterion (8) then
120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (120)
where 119861 is the Brownian bridge
Acknowledgment
The author would like to thank the referees for the carefulreading of the paper and helpful remarks
References
[1] Z Cai and G G Roussas ldquoWeak convergence for smoothestimator of a distribution function under negative associationrdquoStochastic Analysis and Applications vol 17 no 2 pp 145ndash1681999
[2] Z Cai andG G Roussas ldquoEfficient Estimation of a DistributionFunction under Qudrant Dependencerdquo Scandinavian Journal ofStatistics vol 25 no 1 pp 211ndash224 1998
[3] ZW Cai and G G Roussas ldquoUniform strong estimation under120572-mixing with ratesrdquo Statistics amp Probability Letters vol 15 no1 pp 47ndash55 1992
[4] G G Roussas ldquoKernel estimates under association stronguniform consistencyrdquo Statistics amp Probability Letters vol 12 no5 pp 393ndash403 1991
[5] Y F Li and S C Yang ldquoUniformly asymptotic normalityof the smooth estimation of the distribution function underassociated samplesrdquo Acta Mathematicae Applicatae Sinica vol28 no 4 pp 639ndash651 2005
[6] Y Li C Wei and S Yang ldquoThe recursive kernel distributionfunction estimator based onnegatively and positively associatedsequencesrdquo Communications in Statistics vol 39 no 20 pp3585ndash3595 2010
[7] H Yu ldquoA Glivenko-Cantelli lemma and weak convergence forempirical processes of associated sequencesrdquo ProbabilityTheoryand Related Fields vol 95 no 3 pp 357ndash370 1993
[8] Q M Shao and H Yu ldquoWeak convergence for weightedempirical processes of dependent sequencesrdquo The Annals ofProbability vol 24 no 4 pp 2098ndash2127 1996
[9] P Billingsley Convergence of Probability Measures Wiley Seriesin Probability and Statistics Probability and Statistics JohnWiley amp Sons New York NY USA 1999
[10] J D Esary F Proschan and D W Walkup ldquoAssociation ofrandom variables with applicationsrdquo Annals of MathematicalStatistics vol 38 pp 1466ndash1474 1967
[11] A Bulinski and A Shashkin Limit Theorems for AssociatedRandom Fields and Related Systems vol 10 of Advanced Serieson Statistical Science and Applied Probability World Scientific2007
[12] S Karlin and Y Rinott ldquoClasses of orderings of measures andrelated correlation inequalities I Multivariate totally positivedistributionsrdquo Journal of Multivariate Analysis vol 10 no 4 pp467ndash498 1980
[13] P Matuła and M Ziemba ldquoGeneralized covariance inequali-tiesrdquo Central European Journal of Mathematics vol 9 no 2 pp281ndash293 2011
[14] P Doukhan and S Louhichi ldquoA new weak dependence condi-tion and applications to moment inequalitiesrdquo Stochastic Proc-esses and their Applications vol 84 no 2 pp 313ndash342 1999
[15] P Matuła ldquoA note on some inequalities for certain classes ofpositively dependent random variablesrdquo Probability and Math-ematical Statistics vol 24 no 1 pp 17ndash26 2004
[16] S Louhichi ldquoWeak convergence for empirical processes of asso-ciated sequencesrdquo Annales de lrsquoInstitut Henri Poincare vol 36no 5 pp 547ndash567 2000
[17] K Joag-Dev and F Proschan ldquoNegative association of randomvariables with applicationsrdquoThe Annals of Statistics vol 11 no1 pp 286ndash295 1983
[18] C M Newman ldquoAsymptotic independence and limit theoremsfor positively and negatively dependent random variablesrdquo inInequalities in Statistics and Probability Y L Tong Ed vol 5pp 127ndash140 Institute ofMathematical StatisticsHayward CalifUSA 1984
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
ISRN Probability and Statistics 9
It turns out that it suffices to have
1003816100381610038161003816Cov (1198801 119880
1+119899)1003816100381610038161003816 = 119874(
1
1198993119901(119901minus1)) (82)
where 119901 gt 4 is a positive constant taken from the tightnesscriterion (8) It is easy to see that asymptotically we get therate 3
On the way to prove it we will also take use of aRosenthal-type inequality due to Shao andYu (seeTheorem 2in [8]) we shall recall in the following lemma
Lemma 5 Let 119901 gt 2 and 119891 be a real valued function boundedby 1 with bounded first derivative Suppose that 119883
119899119899ge1
is asequence of stationary and associated rvrsquos such that for 119899 isin N
Cov (1198831 119883
119899) = 119874 (119899
minus119887) for some 119887 gt 119901 minus 1 (83)
Then for any 120583 gt 0 there exists some positive constant 119896120583
independent of the function 119891 for which
1198641003816100381610038161003816119878119899
(119891) minus 119864119878119899(119891)
1003816100381610038161003816
119901
le 119896120583(119899
1+12058310038171003817100381710038171003817119891
101584010038171003817100381710038171003817
2
+1198991199012 [
[
119899
sum
119895=1
10038161003816100381610038161003816Cov (119891 (119883
1)119891 (119883
119895))10038161003816100381610038161003816]
]
)
(84)
As we can see the lemma assumes association but itworks for negatively associated rvrsquos as well since in the proofit reaches back the result of Newman (see Proposition 15 in[18]) where both types of association are allowed
Let us recall that 120572119899(119909) = radic119899(119865
119899(119909) minus 119864119865
119899(119909)) where
119865119899(119909) = (1119899)sum
119899
119895=1119870((119909 minus 119883
119895)ℎ
119899) It is easy to see that
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
= 119899minus1199012
11986410038161003816100381610038161003816119878119899(119891 (119883
119895)) minus 119864119878
119899(119891 (119883
119895))10038161003816100381610038161003816
119901
(85)
where
119891 (119883119895) = 119870(
119909 minus 119883119895
ℎ119899
) minus 119870(
119910 minus 119883119895
ℎ119899
)
119878119899(119891) =
119899
sum
119895=1
119891 (119883119895)
(86)
With an intent to use Lemma 5 we need 119891 to be boundedfrom above by 1 (which in our case is obvious) and to havebounded 119891
1015840 thus we assume
sup119905isinR
10038161003816100381610038161003816119870
1015840(119905)
10038161003816100381610038161003816
1
ℎ119899
le119862
119870
ℎ119899
lt infin (87)
Now applying inequality (84) we have
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
le119896120583119899minus1199012
1198991+120583
1198622
119870
ℎ2
119899
+1198991199012[
[
119899
sum
119895=1
10038161003816100381610038161003816Cov (119891 (119883
1)119891 (119883
119895))10038161003816100381610038161003816]
]
1199012
(88)
Newman showed in [18] that
Cov (1198911(119883) 119891
2(119884))
= ∬R2119891
1015840
1(119909) 119891
1015840
2(119910)
times [119875 (119883 le 119909 119884 le 119910) minus 119875 (119883 le 119909) 119875 (119884 le 119910)] 119889119909 119889119910
(89)
if 1198911and 119891
2are real valued functions on R having square
integrable derivatives 1198911015840
1and 119891
1015840
2 respectively and provided
that 1198911(119883) 119891
2(119884) have finite secondmoments In light of that
equation and linearity of covariance we get10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
=
1003816100381610038161003816100381610038161003816∬
R2[119875 (119883
1lt 119909 minus ℎ
119899119904 119883
119895lt 119909 minus ℎ
119899119905)
minus119875 (1198831lt 119909 minus ℎ
119899119904 119883
119895lt 119910 minus ℎ
119899119905)]
minus [119875 (1198831lt 119909 minus ℎ
119899119904) 119875 (119883
119895lt 119909 minus ℎ
119899119905)
minus119875 (1198831lt 119909 minus ℎ
119899119904) 119875 (119883
119895lt 119910 minus ℎ
119899119905)]
minus [119875 (1198831lt 119910 minus ℎ
119899119904 119883
119895lt 119909 minus ℎ
119899119905)
minus119875 (1198831lt 119910 minus ℎ
119899119904 119883
119895lt 119910 minus ℎ
119899119905)]
minus [119875 (1198831lt 119910 minus ℎ
119899119904) 119875 (119883
119895lt 119910 minus ℎ
119899119905)
minus 119875 (1198831lt 119910 minus ℎ
119899119904)
times 119875 (119883119895lt 119909 minus ℎ
119899119905)] 119896 (119904) 119896 (119905) 119889119904 119889119905
10038161003816100381610038161003816100381610038161003816
(90)
Without the loss of generality we may and do assume that119909 gt 119910 thus10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
=
1003816100381610038161003816100381610038161003816∬
R2119875 (119910 minus ℎ
119899119904 le 119883
1lt 119909 minus ℎ
119899119904
119910 minus ℎ119899119905 le 119883
119895lt 119909 minus ℎ
119899119905)
minus 119875 (119910 minus ℎ119899119904 le 119883
1lt 119909 minus ℎ
119899119904)
times 119875 (119910 minus ℎ119899119905 le 119883
119895lt 119909 minus ℎ
119899119905) 119896 (119904) 119896 (119905) 119889119904 119889119905
1003816100381610038161003816100381610038161003816
(91)
and by triangle inequality we get10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
le ∬R2[int
119909minusℎ119899119904
119910minusℎ119899119904
int
119909minusℎ119899119905
119910minusℎ119899119905
1198911198831119883119895(119906 V) 119889119906 119889V
+int
119909minusℎ119899119904
119910minusℎ119899119904
1198911198831(119906) 119889119906int
119909minusℎ119899119905
119910minusℎ119899119905
119891119883119895(V) 119889V] 119896 (119904) 119896 (119905) 119889119904 119889119905
(92)
10 ISRN Probability and Statistics
1198911198831119883119895
(119906 V) 1198911198831
(119906) and 119891119883119895
(V) are the joint pdf of [1198831 119883
119895]
and marginal pdfrsquos of rvrsquos1198831and119883
119895 respectively We need
to further assume that 1198621stands for a common upper bound
of 1198911198831
and 1198911198831119883119895
for all 119895 isin N that is
100381710038171003817100381710038171198911198831
10038171003817100381710038171003817le 119862
1
1003817100381710038171003817100381710038171198911198831119883119895
100381710038171003817100381710038171003817le 119862
1forall119895 ge 1 (93)
Then we finally obtain
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816le 2119862
1
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
where = max 1198621 119862
2
1
(94)
On the other hand proceeding like Cai and Roussas in [1] wecan shortly get
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
le
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119909 minus 119883
1
ℎ119899
) 119870(
119909 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119909 minus 119883
1
ℎ119899
) 119870(
119910 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119910 minus 119883
1
ℎ119899
) 119870(
119909 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119910 minus 119883
1
ℎ119899
) 119870(
119910 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
le 41198622
10038161003816100381610038161003816Cov (119883
1 119883
119895)10038161003816100381610038161003816
13
(95)
where 1198622is a constant relevant to the covariance inequality
for negatively associated rvrsquos (see [13])We now arrive at the following inequality
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
le max 1198621 119862
2
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
10038161003816100381610038161003816Cov(119883
1 119883
119895)10038161003816100381610038161003816
13
(96)
Assuming |Cov(119891(1198831) 119891(119883
119895))| = 119874(119895
minus119903) and proceeding
similarly to (68) we can get
[
[
119899
sum
119895=1
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816]
]
1199012
le max 1198621 119862
2 [
[
sum
119895lt|119909minus119910|minus2(1199033)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
+ sum
119895ge|119909minus119910|minus2(1199033)
1
1198951199033]
]
1199012
le 2max 1198621 119862
21003816100381610038161003816119909 minus 119910
1003816100381610038161003816
((119903minus3)119903)119901
(97)
To sum up we obtain the following inequality
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
le 119896120583119899
1+120583minus1199012119862
2
119870
ℎ2
119899
+ 2max 1198621 119862
21003816100381610038161003816119909 minus 119910
1003816100381610038161003816
((119903minus3)119903)119901
le 1198621198991+120583minus1199012 1
ℎ2
119899
+1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
((119903minus3)119903)119901
(98)
where 119862 = 119896120583max1198622
119896 2max119862
1 119862
2 The formula of the
tightness criterion (8) implies that ((119903 minus 3)119903)119901 gt 1 so
119903 gt3119901
119901 minus 1 where 119901 gt 2 (99)
Let us recall the assumptions imposed on the bandwidhtsℎ
119899119899ge1
by Cai and Roussas in [1]B1 lim
119899rarrinfinℎ119899= 0 and ℎ
119899gt 0 for all 119899 isin N
+
B2 lim119899rarrinfin
119899ℎ119899= infin thus ℎ
119899= 119874(1119899
1minus120573) 120573 gt 0
B3 lim119899rarrinfin
119899ℎ4
119899= 0 hence ℎ
119899= 119874(1119899
14+120575) 120575 gt 0
In light of the above
1198991+120583minus1199012 1
ℎ2
119899
= 11989932+120583+2120575minus1199012
(100)
where 120583 gt 0 119901 gt 2 and 0 lt 120575 lt 34 Confrontation with thetightness criterion (8) forces
3
2+ 120583 + 2120575 minus
119901
2lt minus
1
2 (101)
which implies 119901 gt 4 Let us conclude with the followingtheorem
Theorem6 Let 120572119899(119909) = radic119899(119865
119899(119909)minus119864119865
119899(119909)) be the empirical
process built on the kernel estimator of the df119865 for a stationarysequence of negatively associated rvrsquos Assume that conditionsA1 A2 (87) B1 B2 B3 are satisfied Then provided that thetightness criterion (8) holds with 119901 gt 4 it suffices to demand
1003816100381610038161003816Cov (1198801 119880
1+119899)1003816100381610038161003816 = 119874(
1
1198993119901(119901minus1)) (102)
in order to obtain120572
119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (103)
where 119861(sdot) is the zero mean Gaussian process with covariancestructure defined by
1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)
+
infin
sum
119895=1
[ Cov (119868 [1198831le 119909] 119868 [119883
119895+1le 119910])
+ Cov (119868 [1198831le 119910] 119868 [119883
119895+1le 119909])]
(104)
and 119909 119910 isin [0 1]
Proof The remaining covergence of finite-dimensional dis-tributions of 120572
119899(119909) is established in [1]
ISRN Probability and Statistics 11
5 Weak Convergence of theRecursive Kernel-Type Empirical Processunder IID Assumption
The aim of this section is to show weak convergence of theempirical process
120572119899(119909) =
1
radic119899
119899
sum
119895=1
[119870(
119909 minus 119883119895
ℎ119895
) minus 119864119870(
119909 minus 119883119895
ℎ119895
)] (105)
built on iid rvrsquos 119883119895119895ge1
Wewill prove that120572
119899(sdot) convergesweakly to the Brownian
bridge 119861(sdot) with the following covariance structure
1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910) for 119909 119910 isin [0 1]
(106)
It turns out that the tightness criterion (8) suffices to reachthe goal Convergence of finite-dimensional distributions of120572
119899(sdot) holds and we shall show itProceeding like Cai and Roussas in [1] we need to show
that for any 119886 119887 isin R
119886120572119899(119909) + 119887120572
119899(119910) 997888rarr 119886119861 (119909) + 119887119861 (119910) in distribution
(107)
Let us introduce the notation
119884119894(119909) = 119870(
119909 minus 119883119894
ℎ119894
) minus 119864119870(119909 minus 119883
119894
ℎ119894
) (108)
and look into the covariance structure of 120572119899(sdot)
Cov (120572119899(119909) 120572
119899(119910))
= Cov( 1
radic119899
119899
sum
119894=1
119884119894(119909)
1
radic119899
119899
sum
119894=1
119884119894(119910))
=1
119899
119899
sum
119894=1
Cov (119884119894(119909) 119884
119894(119910))
=1
119899
119899
sum
119894=1
Cov(119870(119909 minus 119883
119894
ℎ119894
) 119870(119910 minus 119883
119894
ℎ119894
))
(109)
where the second equality follows from assumed indepen-dence of rvrsquos 119883
119894119894ge1
Firstly we observe that each summand converges to119865(119909and
119910) minus 119865(119909)119865(119910) since
Cov(119870(119909 minus 119883
119894
ℎ119894
) 119870(119910 minus 119883
119894
ℎ119894
))
= 119864[119870(119909 minus 119883
119894
ℎ119894
)119870(119910 minus 119883
119894
ℎ119894
)]
minus 119864119870(119909 minus 119883
119894
ℎ119894
)119864119870(119910 minus 119883
119894
ℎ119894
)
(110)
where
119864[119870(119909 minus 119883
119894
ℎ119894
)119870(119910 minus 119883
119894
ℎ119894
)]
= int
119909and119910
minusinfin
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
+ int
119909or119910
119909and119910
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
+ int
infin
119909or119910
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
(111)
and 119865 denotes the common df of the rvrsquos 119883119894119894ge1
In [6] itwas shown that
119864119870(119909 minus 119883
119894
ℎ119894
) 997888rarr 119865 (119909) (112)
and recalling that the kernel function 119870 is a df as well wearrive at the conclusion
Secondly applying Toeplitz lemma we get
Cov (120572119899(119909) 120572
119899(119910)) 997888rarr 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)
= 1205902(119909 119910) 119899 997888rarr infin
(113)
Thus
Var (119886120572119899(119909) + 119887120572
119899(119910))
= 1198862 Var (120572
119899(119909)) + 2119886119887Cov (120572
119899(119909) 120572
119899(119910))
+ 1198872 Var (120572
119899(119910))
997888rarr 1198862120590
2(119909 119909) + 2119886119887120590
2(119909 119910) + 119887
2120590
2(119910 119910) 119899 997888rarr infin
(114)
We are now on the way to prove that 119886120572119899(119909) + 119887120572
119899(119910) con-
verges in distribution to 119886119861(119909) + 119887119861(119910) sim N(0 1198862120590
2(119909 119909) +
21198861198871205902(119909 119910) + 119887
2120590
2(119910 119910)) which in other words means that
1
radic119899
infin
sum
119894=1
119886119884119894(119909) + 119887119884
119894(119910)
radicVar (119886120572119899(119909) + 119887120572
119899(119910))
997888rarr N (0 1) (115)
In order to obtain (115) we will use Lyapunov condition
lim119899rarrinfin
1
1198632+120575
119899
119899
sum
119894=1
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884119894(119910)
1003816100381610038161003816
2+120575
= 0 (116)
for 120575 = 1 where 1198632
119899= Var(sum119899
119894=1[119886119884
119894(119909) + 119887119884
119894(119910)]) Since
|119870(sdot) minus 119864119870(sdot)| le 2
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884i (119910)1003816100381610038161003816
3
le 8 (1198863+ 119887
3) (117)
Moreover 1198632
119899= 119899Var(119886120572
119899(119909) + 119887120572
119899(119910)) together with (114)
yields
1198633
119899= 119874 (119899
32) (118)
12 ISRN Probability and Statistics
Combining (117) with (118) we obtain
1
1198633
119899
119899
sum
119894=1
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884119894(119910)
1003816100381610038161003816
3
= 119874 (119899minus12
) (119)
which shows that Lyapunov condition holds and completesthe proof of convergence of finite-dimensional distributionsof 120572
119899(119909)We summarize the result of that section in the following
theorem
Theorem 7 Let 120572119899(119909) be the recursive kernel-type empirical
process defined by (105) built on iid rvrsquos 119883119895119895ge1
withmarginal df 119865 If 120572
119899(sdot) satisfies the tightness criterion (8) then
120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (120)
where 119861 is the Brownian bridge
Acknowledgment
The author would like to thank the referees for the carefulreading of the paper and helpful remarks
References
[1] Z Cai and G G Roussas ldquoWeak convergence for smoothestimator of a distribution function under negative associationrdquoStochastic Analysis and Applications vol 17 no 2 pp 145ndash1681999
[2] Z Cai andG G Roussas ldquoEfficient Estimation of a DistributionFunction under Qudrant Dependencerdquo Scandinavian Journal ofStatistics vol 25 no 1 pp 211ndash224 1998
[3] ZW Cai and G G Roussas ldquoUniform strong estimation under120572-mixing with ratesrdquo Statistics amp Probability Letters vol 15 no1 pp 47ndash55 1992
[4] G G Roussas ldquoKernel estimates under association stronguniform consistencyrdquo Statistics amp Probability Letters vol 12 no5 pp 393ndash403 1991
[5] Y F Li and S C Yang ldquoUniformly asymptotic normalityof the smooth estimation of the distribution function underassociated samplesrdquo Acta Mathematicae Applicatae Sinica vol28 no 4 pp 639ndash651 2005
[6] Y Li C Wei and S Yang ldquoThe recursive kernel distributionfunction estimator based onnegatively and positively associatedsequencesrdquo Communications in Statistics vol 39 no 20 pp3585ndash3595 2010
[7] H Yu ldquoA Glivenko-Cantelli lemma and weak convergence forempirical processes of associated sequencesrdquo ProbabilityTheoryand Related Fields vol 95 no 3 pp 357ndash370 1993
[8] Q M Shao and H Yu ldquoWeak convergence for weightedempirical processes of dependent sequencesrdquo The Annals ofProbability vol 24 no 4 pp 2098ndash2127 1996
[9] P Billingsley Convergence of Probability Measures Wiley Seriesin Probability and Statistics Probability and Statistics JohnWiley amp Sons New York NY USA 1999
[10] J D Esary F Proschan and D W Walkup ldquoAssociation ofrandom variables with applicationsrdquo Annals of MathematicalStatistics vol 38 pp 1466ndash1474 1967
[11] A Bulinski and A Shashkin Limit Theorems for AssociatedRandom Fields and Related Systems vol 10 of Advanced Serieson Statistical Science and Applied Probability World Scientific2007
[12] S Karlin and Y Rinott ldquoClasses of orderings of measures andrelated correlation inequalities I Multivariate totally positivedistributionsrdquo Journal of Multivariate Analysis vol 10 no 4 pp467ndash498 1980
[13] P Matuła and M Ziemba ldquoGeneralized covariance inequali-tiesrdquo Central European Journal of Mathematics vol 9 no 2 pp281ndash293 2011
[14] P Doukhan and S Louhichi ldquoA new weak dependence condi-tion and applications to moment inequalitiesrdquo Stochastic Proc-esses and their Applications vol 84 no 2 pp 313ndash342 1999
[15] P Matuła ldquoA note on some inequalities for certain classes ofpositively dependent random variablesrdquo Probability and Math-ematical Statistics vol 24 no 1 pp 17ndash26 2004
[16] S Louhichi ldquoWeak convergence for empirical processes of asso-ciated sequencesrdquo Annales de lrsquoInstitut Henri Poincare vol 36no 5 pp 547ndash567 2000
[17] K Joag-Dev and F Proschan ldquoNegative association of randomvariables with applicationsrdquoThe Annals of Statistics vol 11 no1 pp 286ndash295 1983
[18] C M Newman ldquoAsymptotic independence and limit theoremsfor positively and negatively dependent random variablesrdquo inInequalities in Statistics and Probability Y L Tong Ed vol 5pp 127ndash140 Institute ofMathematical StatisticsHayward CalifUSA 1984
Submit your manuscripts athttpwwwhindawicom
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Discrete Dynamics in Nature and Society
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Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 ISRN Probability and Statistics
1198911198831119883119895
(119906 V) 1198911198831
(119906) and 119891119883119895
(V) are the joint pdf of [1198831 119883
119895]
and marginal pdfrsquos of rvrsquos1198831and119883
119895 respectively We need
to further assume that 1198621stands for a common upper bound
of 1198911198831
and 1198911198831119883119895
for all 119895 isin N that is
100381710038171003817100381710038171198911198831
10038171003817100381710038171003817le 119862
1
1003817100381710038171003817100381710038171198911198831119883119895
100381710038171003817100381710038171003817le 119862
1forall119895 ge 1 (93)
Then we finally obtain
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816le 2119862
1
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
where = max 1198621 119862
2
1
(94)
On the other hand proceeding like Cai and Roussas in [1] wecan shortly get
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
le
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119909 minus 119883
1
ℎ119899
) 119870(
119909 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119909 minus 119883
1
ℎ119899
) 119870(
119910 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119910 minus 119883
1
ℎ119899
) 119870(
119909 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
Cov(119870(119910 minus 119883
1
ℎ119899
) 119870(
119910 minus 119883119895
ℎ119899
))
100381610038161003816100381610038161003816100381610038161003816
le 41198622
10038161003816100381610038161003816Cov (119883
1 119883
119895)10038161003816100381610038161003816
13
(95)
where 1198622is a constant relevant to the covariance inequality
for negatively associated rvrsquos (see [13])We now arrive at the following inequality
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816
le max 1198621 119862
2
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
10038161003816100381610038161003816Cov(119883
1 119883
119895)10038161003816100381610038161003816
13
(96)
Assuming |Cov(119891(1198831) 119891(119883
119895))| = 119874(119895
minus119903) and proceeding
similarly to (68) we can get
[
[
119899
sum
119895=1
10038161003816100381610038161003816Cov (119891 (119883
1) 119891 (119883
119895))10038161003816100381610038161003816]
]
1199012
le max 1198621 119862
2 [
[
sum
119895lt|119909minus119910|minus2(1199033)
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
+ sum
119895ge|119909minus119910|minus2(1199033)
1
1198951199033]
]
1199012
le 2max 1198621 119862
21003816100381610038161003816119909 minus 119910
1003816100381610038161003816
((119903minus3)119903)119901
(97)
To sum up we obtain the following inequality
1198641003816100381610038161003816120572119899
(119909) minus 120572119899(119910)
1003816100381610038161003816
119901
le 119896120583119899
1+120583minus1199012119862
2
119870
ℎ2
119899
+ 2max 1198621 119862
21003816100381610038161003816119909 minus 119910
1003816100381610038161003816
((119903minus3)119903)119901
le 1198621198991+120583minus1199012 1
ℎ2
119899
+1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
((119903minus3)119903)119901
(98)
where 119862 = 119896120583max1198622
119896 2max119862
1 119862
2 The formula of the
tightness criterion (8) implies that ((119903 minus 3)119903)119901 gt 1 so
119903 gt3119901
119901 minus 1 where 119901 gt 2 (99)
Let us recall the assumptions imposed on the bandwidhtsℎ
119899119899ge1
by Cai and Roussas in [1]B1 lim
119899rarrinfinℎ119899= 0 and ℎ
119899gt 0 for all 119899 isin N
+
B2 lim119899rarrinfin
119899ℎ119899= infin thus ℎ
119899= 119874(1119899
1minus120573) 120573 gt 0
B3 lim119899rarrinfin
119899ℎ4
119899= 0 hence ℎ
119899= 119874(1119899
14+120575) 120575 gt 0
In light of the above
1198991+120583minus1199012 1
ℎ2
119899
= 11989932+120583+2120575minus1199012
(100)
where 120583 gt 0 119901 gt 2 and 0 lt 120575 lt 34 Confrontation with thetightness criterion (8) forces
3
2+ 120583 + 2120575 minus
119901
2lt minus
1
2 (101)
which implies 119901 gt 4 Let us conclude with the followingtheorem
Theorem6 Let 120572119899(119909) = radic119899(119865
119899(119909)minus119864119865
119899(119909)) be the empirical
process built on the kernel estimator of the df119865 for a stationarysequence of negatively associated rvrsquos Assume that conditionsA1 A2 (87) B1 B2 B3 are satisfied Then provided that thetightness criterion (8) holds with 119901 gt 4 it suffices to demand
1003816100381610038161003816Cov (1198801 119880
1+119899)1003816100381610038161003816 = 119874(
1
1198993119901(119901minus1)) (102)
in order to obtain120572
119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (103)
where 119861(sdot) is the zero mean Gaussian process with covariancestructure defined by
1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)
+
infin
sum
119895=1
[ Cov (119868 [1198831le 119909] 119868 [119883
119895+1le 119910])
+ Cov (119868 [1198831le 119910] 119868 [119883
119895+1le 119909])]
(104)
and 119909 119910 isin [0 1]
Proof The remaining covergence of finite-dimensional dis-tributions of 120572
119899(119909) is established in [1]
ISRN Probability and Statistics 11
5 Weak Convergence of theRecursive Kernel-Type Empirical Processunder IID Assumption
The aim of this section is to show weak convergence of theempirical process
120572119899(119909) =
1
radic119899
119899
sum
119895=1
[119870(
119909 minus 119883119895
ℎ119895
) minus 119864119870(
119909 minus 119883119895
ℎ119895
)] (105)
built on iid rvrsquos 119883119895119895ge1
Wewill prove that120572
119899(sdot) convergesweakly to the Brownian
bridge 119861(sdot) with the following covariance structure
1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910) for 119909 119910 isin [0 1]
(106)
It turns out that the tightness criterion (8) suffices to reachthe goal Convergence of finite-dimensional distributions of120572
119899(sdot) holds and we shall show itProceeding like Cai and Roussas in [1] we need to show
that for any 119886 119887 isin R
119886120572119899(119909) + 119887120572
119899(119910) 997888rarr 119886119861 (119909) + 119887119861 (119910) in distribution
(107)
Let us introduce the notation
119884119894(119909) = 119870(
119909 minus 119883119894
ℎ119894
) minus 119864119870(119909 minus 119883
119894
ℎ119894
) (108)
and look into the covariance structure of 120572119899(sdot)
Cov (120572119899(119909) 120572
119899(119910))
= Cov( 1
radic119899
119899
sum
119894=1
119884119894(119909)
1
radic119899
119899
sum
119894=1
119884119894(119910))
=1
119899
119899
sum
119894=1
Cov (119884119894(119909) 119884
119894(119910))
=1
119899
119899
sum
119894=1
Cov(119870(119909 minus 119883
119894
ℎ119894
) 119870(119910 minus 119883
119894
ℎ119894
))
(109)
where the second equality follows from assumed indepen-dence of rvrsquos 119883
119894119894ge1
Firstly we observe that each summand converges to119865(119909and
119910) minus 119865(119909)119865(119910) since
Cov(119870(119909 minus 119883
119894
ℎ119894
) 119870(119910 minus 119883
119894
ℎ119894
))
= 119864[119870(119909 minus 119883
119894
ℎ119894
)119870(119910 minus 119883
119894
ℎ119894
)]
minus 119864119870(119909 minus 119883
119894
ℎ119894
)119864119870(119910 minus 119883
119894
ℎ119894
)
(110)
where
119864[119870(119909 minus 119883
119894
ℎ119894
)119870(119910 minus 119883
119894
ℎ119894
)]
= int
119909and119910
minusinfin
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
+ int
119909or119910
119909and119910
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
+ int
infin
119909or119910
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
(111)
and 119865 denotes the common df of the rvrsquos 119883119894119894ge1
In [6] itwas shown that
119864119870(119909 minus 119883
119894
ℎ119894
) 997888rarr 119865 (119909) (112)
and recalling that the kernel function 119870 is a df as well wearrive at the conclusion
Secondly applying Toeplitz lemma we get
Cov (120572119899(119909) 120572
119899(119910)) 997888rarr 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)
= 1205902(119909 119910) 119899 997888rarr infin
(113)
Thus
Var (119886120572119899(119909) + 119887120572
119899(119910))
= 1198862 Var (120572
119899(119909)) + 2119886119887Cov (120572
119899(119909) 120572
119899(119910))
+ 1198872 Var (120572
119899(119910))
997888rarr 1198862120590
2(119909 119909) + 2119886119887120590
2(119909 119910) + 119887
2120590
2(119910 119910) 119899 997888rarr infin
(114)
We are now on the way to prove that 119886120572119899(119909) + 119887120572
119899(119910) con-
verges in distribution to 119886119861(119909) + 119887119861(119910) sim N(0 1198862120590
2(119909 119909) +
21198861198871205902(119909 119910) + 119887
2120590
2(119910 119910)) which in other words means that
1
radic119899
infin
sum
119894=1
119886119884119894(119909) + 119887119884
119894(119910)
radicVar (119886120572119899(119909) + 119887120572
119899(119910))
997888rarr N (0 1) (115)
In order to obtain (115) we will use Lyapunov condition
lim119899rarrinfin
1
1198632+120575
119899
119899
sum
119894=1
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884119894(119910)
1003816100381610038161003816
2+120575
= 0 (116)
for 120575 = 1 where 1198632
119899= Var(sum119899
119894=1[119886119884
119894(119909) + 119887119884
119894(119910)]) Since
|119870(sdot) minus 119864119870(sdot)| le 2
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884i (119910)1003816100381610038161003816
3
le 8 (1198863+ 119887
3) (117)
Moreover 1198632
119899= 119899Var(119886120572
119899(119909) + 119887120572
119899(119910)) together with (114)
yields
1198633
119899= 119874 (119899
32) (118)
12 ISRN Probability and Statistics
Combining (117) with (118) we obtain
1
1198633
119899
119899
sum
119894=1
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884119894(119910)
1003816100381610038161003816
3
= 119874 (119899minus12
) (119)
which shows that Lyapunov condition holds and completesthe proof of convergence of finite-dimensional distributionsof 120572
119899(119909)We summarize the result of that section in the following
theorem
Theorem 7 Let 120572119899(119909) be the recursive kernel-type empirical
process defined by (105) built on iid rvrsquos 119883119895119895ge1
withmarginal df 119865 If 120572
119899(sdot) satisfies the tightness criterion (8) then
120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (120)
where 119861 is the Brownian bridge
Acknowledgment
The author would like to thank the referees for the carefulreading of the paper and helpful remarks
References
[1] Z Cai and G G Roussas ldquoWeak convergence for smoothestimator of a distribution function under negative associationrdquoStochastic Analysis and Applications vol 17 no 2 pp 145ndash1681999
[2] Z Cai andG G Roussas ldquoEfficient Estimation of a DistributionFunction under Qudrant Dependencerdquo Scandinavian Journal ofStatistics vol 25 no 1 pp 211ndash224 1998
[3] ZW Cai and G G Roussas ldquoUniform strong estimation under120572-mixing with ratesrdquo Statistics amp Probability Letters vol 15 no1 pp 47ndash55 1992
[4] G G Roussas ldquoKernel estimates under association stronguniform consistencyrdquo Statistics amp Probability Letters vol 12 no5 pp 393ndash403 1991
[5] Y F Li and S C Yang ldquoUniformly asymptotic normalityof the smooth estimation of the distribution function underassociated samplesrdquo Acta Mathematicae Applicatae Sinica vol28 no 4 pp 639ndash651 2005
[6] Y Li C Wei and S Yang ldquoThe recursive kernel distributionfunction estimator based onnegatively and positively associatedsequencesrdquo Communications in Statistics vol 39 no 20 pp3585ndash3595 2010
[7] H Yu ldquoA Glivenko-Cantelli lemma and weak convergence forempirical processes of associated sequencesrdquo ProbabilityTheoryand Related Fields vol 95 no 3 pp 357ndash370 1993
[8] Q M Shao and H Yu ldquoWeak convergence for weightedempirical processes of dependent sequencesrdquo The Annals ofProbability vol 24 no 4 pp 2098ndash2127 1996
[9] P Billingsley Convergence of Probability Measures Wiley Seriesin Probability and Statistics Probability and Statistics JohnWiley amp Sons New York NY USA 1999
[10] J D Esary F Proschan and D W Walkup ldquoAssociation ofrandom variables with applicationsrdquo Annals of MathematicalStatistics vol 38 pp 1466ndash1474 1967
[11] A Bulinski and A Shashkin Limit Theorems for AssociatedRandom Fields and Related Systems vol 10 of Advanced Serieson Statistical Science and Applied Probability World Scientific2007
[12] S Karlin and Y Rinott ldquoClasses of orderings of measures andrelated correlation inequalities I Multivariate totally positivedistributionsrdquo Journal of Multivariate Analysis vol 10 no 4 pp467ndash498 1980
[13] P Matuła and M Ziemba ldquoGeneralized covariance inequali-tiesrdquo Central European Journal of Mathematics vol 9 no 2 pp281ndash293 2011
[14] P Doukhan and S Louhichi ldquoA new weak dependence condi-tion and applications to moment inequalitiesrdquo Stochastic Proc-esses and their Applications vol 84 no 2 pp 313ndash342 1999
[15] P Matuła ldquoA note on some inequalities for certain classes ofpositively dependent random variablesrdquo Probability and Math-ematical Statistics vol 24 no 1 pp 17ndash26 2004
[16] S Louhichi ldquoWeak convergence for empirical processes of asso-ciated sequencesrdquo Annales de lrsquoInstitut Henri Poincare vol 36no 5 pp 547ndash567 2000
[17] K Joag-Dev and F Proschan ldquoNegative association of randomvariables with applicationsrdquoThe Annals of Statistics vol 11 no1 pp 286ndash295 1983
[18] C M Newman ldquoAsymptotic independence and limit theoremsfor positively and negatively dependent random variablesrdquo inInequalities in Statistics and Probability Y L Tong Ed vol 5pp 127ndash140 Institute ofMathematical StatisticsHayward CalifUSA 1984
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
ISRN Probability and Statistics 11
5 Weak Convergence of theRecursive Kernel-Type Empirical Processunder IID Assumption
The aim of this section is to show weak convergence of theempirical process
120572119899(119909) =
1
radic119899
119899
sum
119895=1
[119870(
119909 minus 119883119895
ℎ119895
) minus 119864119870(
119909 minus 119883119895
ℎ119895
)] (105)
built on iid rvrsquos 119883119895119895ge1
Wewill prove that120572
119899(sdot) convergesweakly to the Brownian
bridge 119861(sdot) with the following covariance structure
1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910) for 119909 119910 isin [0 1]
(106)
It turns out that the tightness criterion (8) suffices to reachthe goal Convergence of finite-dimensional distributions of120572
119899(sdot) holds and we shall show itProceeding like Cai and Roussas in [1] we need to show
that for any 119886 119887 isin R
119886120572119899(119909) + 119887120572
119899(119910) 997888rarr 119886119861 (119909) + 119887119861 (119910) in distribution
(107)
Let us introduce the notation
119884119894(119909) = 119870(
119909 minus 119883119894
ℎ119894
) minus 119864119870(119909 minus 119883
119894
ℎ119894
) (108)
and look into the covariance structure of 120572119899(sdot)
Cov (120572119899(119909) 120572
119899(119910))
= Cov( 1
radic119899
119899
sum
119894=1
119884119894(119909)
1
radic119899
119899
sum
119894=1
119884119894(119910))
=1
119899
119899
sum
119894=1
Cov (119884119894(119909) 119884
119894(119910))
=1
119899
119899
sum
119894=1
Cov(119870(119909 minus 119883
119894
ℎ119894
) 119870(119910 minus 119883
119894
ℎ119894
))
(109)
where the second equality follows from assumed indepen-dence of rvrsquos 119883
119894119894ge1
Firstly we observe that each summand converges to119865(119909and
119910) minus 119865(119909)119865(119910) since
Cov(119870(119909 minus 119883
119894
ℎ119894
) 119870(119910 minus 119883
119894
ℎ119894
))
= 119864[119870(119909 minus 119883
119894
ℎ119894
)119870(119910 minus 119883
119894
ℎ119894
)]
minus 119864119870(119909 minus 119883
119894
ℎ119894
)119864119870(119910 minus 119883
119894
ℎ119894
)
(110)
where
119864[119870(119909 minus 119883
119894
ℎ119894
)119870(119910 minus 119883
119894
ℎ119894
)]
= int
119909and119910
minusinfin
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
+ int
119909or119910
119909and119910
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
+ int
infin
119909or119910
119870(119909 minus 119905
ℎ119894
)119870(119910 minus 119905
ℎ119894
)119889119865 (119905)
(111)
and 119865 denotes the common df of the rvrsquos 119883119894119894ge1
In [6] itwas shown that
119864119870(119909 minus 119883
119894
ℎ119894
) 997888rarr 119865 (119909) (112)
and recalling that the kernel function 119870 is a df as well wearrive at the conclusion
Secondly applying Toeplitz lemma we get
Cov (120572119899(119909) 120572
119899(119910)) 997888rarr 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)
= 1205902(119909 119910) 119899 997888rarr infin
(113)
Thus
Var (119886120572119899(119909) + 119887120572
119899(119910))
= 1198862 Var (120572
119899(119909)) + 2119886119887Cov (120572
119899(119909) 120572
119899(119910))
+ 1198872 Var (120572
119899(119910))
997888rarr 1198862120590
2(119909 119909) + 2119886119887120590
2(119909 119910) + 119887
2120590
2(119910 119910) 119899 997888rarr infin
(114)
We are now on the way to prove that 119886120572119899(119909) + 119887120572
119899(119910) con-
verges in distribution to 119886119861(119909) + 119887119861(119910) sim N(0 1198862120590
2(119909 119909) +
21198861198871205902(119909 119910) + 119887
2120590
2(119910 119910)) which in other words means that
1
radic119899
infin
sum
119894=1
119886119884119894(119909) + 119887119884
119894(119910)
radicVar (119886120572119899(119909) + 119887120572
119899(119910))
997888rarr N (0 1) (115)
In order to obtain (115) we will use Lyapunov condition
lim119899rarrinfin
1
1198632+120575
119899
119899
sum
119894=1
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884119894(119910)
1003816100381610038161003816
2+120575
= 0 (116)
for 120575 = 1 where 1198632
119899= Var(sum119899
119894=1[119886119884
119894(119909) + 119887119884
119894(119910)]) Since
|119870(sdot) minus 119864119870(sdot)| le 2
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884i (119910)1003816100381610038161003816
3
le 8 (1198863+ 119887
3) (117)
Moreover 1198632
119899= 119899Var(119886120572
119899(119909) + 119887120572
119899(119910)) together with (114)
yields
1198633
119899= 119874 (119899
32) (118)
12 ISRN Probability and Statistics
Combining (117) with (118) we obtain
1
1198633
119899
119899
sum
119894=1
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884119894(119910)
1003816100381610038161003816
3
= 119874 (119899minus12
) (119)
which shows that Lyapunov condition holds and completesthe proof of convergence of finite-dimensional distributionsof 120572
119899(119909)We summarize the result of that section in the following
theorem
Theorem 7 Let 120572119899(119909) be the recursive kernel-type empirical
process defined by (105) built on iid rvrsquos 119883119895119895ge1
withmarginal df 119865 If 120572
119899(sdot) satisfies the tightness criterion (8) then
120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (120)
where 119861 is the Brownian bridge
Acknowledgment
The author would like to thank the referees for the carefulreading of the paper and helpful remarks
References
[1] Z Cai and G G Roussas ldquoWeak convergence for smoothestimator of a distribution function under negative associationrdquoStochastic Analysis and Applications vol 17 no 2 pp 145ndash1681999
[2] Z Cai andG G Roussas ldquoEfficient Estimation of a DistributionFunction under Qudrant Dependencerdquo Scandinavian Journal ofStatistics vol 25 no 1 pp 211ndash224 1998
[3] ZW Cai and G G Roussas ldquoUniform strong estimation under120572-mixing with ratesrdquo Statistics amp Probability Letters vol 15 no1 pp 47ndash55 1992
[4] G G Roussas ldquoKernel estimates under association stronguniform consistencyrdquo Statistics amp Probability Letters vol 12 no5 pp 393ndash403 1991
[5] Y F Li and S C Yang ldquoUniformly asymptotic normalityof the smooth estimation of the distribution function underassociated samplesrdquo Acta Mathematicae Applicatae Sinica vol28 no 4 pp 639ndash651 2005
[6] Y Li C Wei and S Yang ldquoThe recursive kernel distributionfunction estimator based onnegatively and positively associatedsequencesrdquo Communications in Statistics vol 39 no 20 pp3585ndash3595 2010
[7] H Yu ldquoA Glivenko-Cantelli lemma and weak convergence forempirical processes of associated sequencesrdquo ProbabilityTheoryand Related Fields vol 95 no 3 pp 357ndash370 1993
[8] Q M Shao and H Yu ldquoWeak convergence for weightedempirical processes of dependent sequencesrdquo The Annals ofProbability vol 24 no 4 pp 2098ndash2127 1996
[9] P Billingsley Convergence of Probability Measures Wiley Seriesin Probability and Statistics Probability and Statistics JohnWiley amp Sons New York NY USA 1999
[10] J D Esary F Proschan and D W Walkup ldquoAssociation ofrandom variables with applicationsrdquo Annals of MathematicalStatistics vol 38 pp 1466ndash1474 1967
[11] A Bulinski and A Shashkin Limit Theorems for AssociatedRandom Fields and Related Systems vol 10 of Advanced Serieson Statistical Science and Applied Probability World Scientific2007
[12] S Karlin and Y Rinott ldquoClasses of orderings of measures andrelated correlation inequalities I Multivariate totally positivedistributionsrdquo Journal of Multivariate Analysis vol 10 no 4 pp467ndash498 1980
[13] P Matuła and M Ziemba ldquoGeneralized covariance inequali-tiesrdquo Central European Journal of Mathematics vol 9 no 2 pp281ndash293 2011
[14] P Doukhan and S Louhichi ldquoA new weak dependence condi-tion and applications to moment inequalitiesrdquo Stochastic Proc-esses and their Applications vol 84 no 2 pp 313ndash342 1999
[15] P Matuła ldquoA note on some inequalities for certain classes ofpositively dependent random variablesrdquo Probability and Math-ematical Statistics vol 24 no 1 pp 17ndash26 2004
[16] S Louhichi ldquoWeak convergence for empirical processes of asso-ciated sequencesrdquo Annales de lrsquoInstitut Henri Poincare vol 36no 5 pp 547ndash567 2000
[17] K Joag-Dev and F Proschan ldquoNegative association of randomvariables with applicationsrdquoThe Annals of Statistics vol 11 no1 pp 286ndash295 1983
[18] C M Newman ldquoAsymptotic independence and limit theoremsfor positively and negatively dependent random variablesrdquo inInequalities in Statistics and Probability Y L Tong Ed vol 5pp 127ndash140 Institute ofMathematical StatisticsHayward CalifUSA 1984
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 ISRN Probability and Statistics
Combining (117) with (118) we obtain
1
1198633
119899
119899
sum
119894=1
1198641003816100381610038161003816119886119884119894
(119909) + 119887119884119894(119910)
1003816100381610038161003816
3
= 119874 (119899minus12
) (119)
which shows that Lyapunov condition holds and completesthe proof of convergence of finite-dimensional distributionsof 120572
119899(119909)We summarize the result of that section in the following
theorem
Theorem 7 Let 120572119899(119909) be the recursive kernel-type empirical
process defined by (105) built on iid rvrsquos 119883119895119895ge1
withmarginal df 119865 If 120572
119899(sdot) satisfies the tightness criterion (8) then
120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (120)
where 119861 is the Brownian bridge
Acknowledgment
The author would like to thank the referees for the carefulreading of the paper and helpful remarks
References
[1] Z Cai and G G Roussas ldquoWeak convergence for smoothestimator of a distribution function under negative associationrdquoStochastic Analysis and Applications vol 17 no 2 pp 145ndash1681999
[2] Z Cai andG G Roussas ldquoEfficient Estimation of a DistributionFunction under Qudrant Dependencerdquo Scandinavian Journal ofStatistics vol 25 no 1 pp 211ndash224 1998
[3] ZW Cai and G G Roussas ldquoUniform strong estimation under120572-mixing with ratesrdquo Statistics amp Probability Letters vol 15 no1 pp 47ndash55 1992
[4] G G Roussas ldquoKernel estimates under association stronguniform consistencyrdquo Statistics amp Probability Letters vol 12 no5 pp 393ndash403 1991
[5] Y F Li and S C Yang ldquoUniformly asymptotic normalityof the smooth estimation of the distribution function underassociated samplesrdquo Acta Mathematicae Applicatae Sinica vol28 no 4 pp 639ndash651 2005
[6] Y Li C Wei and S Yang ldquoThe recursive kernel distributionfunction estimator based onnegatively and positively associatedsequencesrdquo Communications in Statistics vol 39 no 20 pp3585ndash3595 2010
[7] H Yu ldquoA Glivenko-Cantelli lemma and weak convergence forempirical processes of associated sequencesrdquo ProbabilityTheoryand Related Fields vol 95 no 3 pp 357ndash370 1993
[8] Q M Shao and H Yu ldquoWeak convergence for weightedempirical processes of dependent sequencesrdquo The Annals ofProbability vol 24 no 4 pp 2098ndash2127 1996
[9] P Billingsley Convergence of Probability Measures Wiley Seriesin Probability and Statistics Probability and Statistics JohnWiley amp Sons New York NY USA 1999
[10] J D Esary F Proschan and D W Walkup ldquoAssociation ofrandom variables with applicationsrdquo Annals of MathematicalStatistics vol 38 pp 1466ndash1474 1967
[11] A Bulinski and A Shashkin Limit Theorems for AssociatedRandom Fields and Related Systems vol 10 of Advanced Serieson Statistical Science and Applied Probability World Scientific2007
[12] S Karlin and Y Rinott ldquoClasses of orderings of measures andrelated correlation inequalities I Multivariate totally positivedistributionsrdquo Journal of Multivariate Analysis vol 10 no 4 pp467ndash498 1980
[13] P Matuła and M Ziemba ldquoGeneralized covariance inequali-tiesrdquo Central European Journal of Mathematics vol 9 no 2 pp281ndash293 2011
[14] P Doukhan and S Louhichi ldquoA new weak dependence condi-tion and applications to moment inequalitiesrdquo Stochastic Proc-esses and their Applications vol 84 no 2 pp 313ndash342 1999
[15] P Matuła ldquoA note on some inequalities for certain classes ofpositively dependent random variablesrdquo Probability and Math-ematical Statistics vol 24 no 1 pp 17ndash26 2004
[16] S Louhichi ldquoWeak convergence for empirical processes of asso-ciated sequencesrdquo Annales de lrsquoInstitut Henri Poincare vol 36no 5 pp 547ndash567 2000
[17] K Joag-Dev and F Proschan ldquoNegative association of randomvariables with applicationsrdquoThe Annals of Statistics vol 11 no1 pp 286ndash295 1983
[18] C M Newman ldquoAsymptotic independence and limit theoremsfor positively and negatively dependent random variablesrdquo inInequalities in Statistics and Probability Y L Tong Ed vol 5pp 127ndash140 Institute ofMathematical StatisticsHayward CalifUSA 1984
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of