on the asymptotically uniform distribution modulo 1 of ...the concept of asymptotically uniform...
TRANSCRIPT
On the asymptotically uniform distribution modulo 1 of extremeorder statisticsCitation for published version (APA):Brands, J. J. A. M., & Wilms, R. J. G. (1992). On the asymptotically uniform distribution modulo 1 of extremeorder statistics. (Revised ed.) (Memorandum COSOR; Vol. 9138). Technische Universiteit Eindhoven.
Document status and date:Published: 01/01/1992
Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)
Please check the document version of this publication:
• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publication
General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.
• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.
If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverne
Take down policyIf you believe that this document breaches copyright please contact us at:[email protected] details and we will investigate your claim.
Download date: 15. Oct. 2020
EINDHOVEN UNIVERSITY OF TECHNOLOGYDepartment of Mathematics and Computing Science
Memorandum CaSOR 91-38(revised version)
On the asymptotically uniform distributionmodulo 1 of extreme order stastistics
R.J.G. WilmsJ.J.A.M. Brands
Eindhoven, December 1991Revised June 1992
Revised January 1993The Netherlands
Eindhoven University of TechnologyDepartment of Mathematics and Computing ScienceProbability theory, statistics, operations research and systems theoryP.O. Box 5135600 MB Eindhoven - The Netherlands
Secretariate: Dommelbuilding 0.03Telephone: 040-47 3130
ISSN 0926 4493
ON THE ASYMPTOTICALLY UNIFORM DISTRIBUTION MODULO 1
OF EX'l'REME OImER STATISTICS
R.J.G. Wilms
and
J.J.A.M. Brands
Eindhoven University of Technology
Department of Mathematics and Computing Science
P.O. Box 513
5600 MB Eindhoven
The Netherlands
Abstract
00Let (X) be a sequence of independent and identicallym 1
distributed random variables. We give sufficient
conditions for the fractional part of max(X , ... ,X ) to1 n
converge in distribution, as n~, to a random variable
with a uniform distribution on [0,1).
Key Words & Phrases: distribution (modulo 1), Fourier-
Stieltjes coefficients, fractional part.
1
1. Introduction and notation.
The concept of asymptotically uniform distribution (or
equidistribution) modulo 1 (mod 1) of a sequence is well known in
number theory, and is also important in random number generation
(Ripley (1987». The celebrated Weyl criterion states a necessary
coand sufficient condition for a sequence (x) of real numbers to
n 1
be asymptotically uniformly distributed (mod 1) (see Kuipers and
Niederreiter (1974». Holewijn (1969) generalizes this criterion
to a criterion for a sequence (X)CO of independent randomn 1
variables to be uniformly distributed (mod 1) almost surely.
Let {X} denote the fractional part of a random variable (r.v.) X,
defined by {X} :=X- [Xl, where [Xl denotes the integer part of X,
cothe largest integer not exceeding X. A sequence (Z) of r.v.'s is
n 1
said to be asymptotically uniform in distribution (mod 1) if
(1) (n~) ,
where U is uniformly distributed on [0,1). It is well known and
easily proved (see e.g. Schatte (1983» that (1) holds whenn
Z =\ X where the X are independent and identically distributedn L m' m
1
(LLd.)n
and non-lattice. For Z =\ ~X relationnLmm
1
(1) does not
generally hold; Jagers (1990), solving a problem by Steutel, shows
that (1) does not hold for exponentionally distributed X . Since,m
n 1 din this case \ -X = max (X , ... ,X ), it is of interest to considerL mmIn
1
max(X , ... ,X ) in general.1 n
The object of this paper is to give sufficient conditions for
(Z ) = (max (X , ... , X » to be asymptotically uniform in distributionn 1 n
(mod 1). Clearly, if the right endpoint of distribution function
2
(d.f.) F, W(F) :=sup{x: F(x)<l}, is finite, then {Z } convergesn
almost surely to {W(F)}. Therefore, we shall assume that W(F) is
infinite.
Throughout this paper,d
X=X1
,X2
' ••• will be a sequence of Ll.d.
Inr.v.'s with right-continuous d.f. F=FX
on ~ with w(F)=~.
addition, we denote: Z =max (X , ... , X ); U is the r. v. with uniformn 1 n
distribution on [0, 1) , and V is the r . v . with standard
exponentional distribution, Le. V-Exp(l). Further we use the
following abbreviations:
n={h:~+~; h is piecewise continuous and non-decreasing},
t-xG (x) =exp (-e ) ,
t
t-x t-xg (x) =G' (x) =e exp (-e )
t t
13k
(x) =exp (21tikx) (xe~, kel),
x+=maX(x,O) (xe~) .
We represent F in terms of hazard functions: For any X we can
write
(2) X ~ h (V) ,
for some non-decreasing function h :~+~. The right-continuous
(generalized) inverse function of h, defined by
h(x)=-log(l-F(x», is the cumulative hazard function of F; when F
has a derivative F', then h' (x)=F' (x)/(l-F(x» is called the
hazard rate of F.
In section 2 we give some properties of Fourier-Stieltjes
ofthe F.S.S.Le.
sequences (F. S. S.' s). In section 3 we prClve the main result of
the condition h «Y+logn) )~,+ .this paper:
h ( (Y+logn) +) tends to zero as n~, is necessary and sufficient for
dis with{Z }~U. Here Y a r.v. d.f. Go' one of the three
n
possible types of extreme value distributions (see Resnick
3
(1987) ) • In addition, the requirement that
(t~) is sufficient for this condition. This leads to a rather
simple sufficient condition: if F has a hazard rate that tends
monotonically to zero, then {Z }~u (n~). In section 4 we given
some examples and briefly consider the connection with extreme
value theory.
2. Properties of Fourier-Stieltjes Sequences.
We start by giving some notations and definitions. Let '§ [0,1)
denote the class of right-continuous d.f.'s on ~ with support in
[0,1], and F(1-0)=1. We recall the definition of the F.S.S. of
such d.f.'s.
Definition 1. Let F be a d.f.
of F is defined by
J 2nikxcF(k)= e dF(x)
[0, 1)
(kel) .
•
andClearly cF
(O)=l,instead of cF
•
X
(kel). Further we have Cu(k)=O for k~O. Since
We
exp(2nik{x})=exp(2nikx) (xe~), we have for any X the trivial but
useful identity
Next, we state the uniqueness and the continuity theorems for
4
F.S.S.'s of d.f.'s in ~[O,l). For the proofs we refer to Zygmund
(1968) •
Proposition 1. Let F,Ge~[O,l). If cF=cG
' then F(x)=G(x) (xeR). c
00Proposition 2. Let (F) be a sequence of d.f.'s in ~[0,1) and letn 1
(c ) 00 be the corresponding sequence of F. S. S. ' s. The sequencen 1
(F )00 converges weakly to a d.f. Fe~[O,l) iff c (k)~(k) for keln 1 n
(n~). This sequence c then is the F.S.S. of F. c
The foregoing propositions justify the following definition of
asymptotic uniformity (mod 1) in distribution.
Definition 2. A sequence (y)OO of r.v.'s is said to ben 1
asymptotically uniform in distribution (a.u.d.) (mod 1) if
(n~) ,
or equivalently, if
fi~ c {Y } (k) =0n
(k:¢O) .
•
3. Convergence of the sequence (z }.n
In this section we give rather weak sufficient conditions on F for
(Z ) to be a.u.d. (mod 1). We now state the main result.n
dTheorem 1. Let X = h(V) for some non-decreasing function h:R+~,
and V-Exp(l). Then
5
(3)IX)
lim J~k(h(s+logn»g (s)dsn~ 0
-logn
o (k:;tO)
iff {Z }~U (n~).n
Proof: We first note that
{Z }={max(X, ... ,X ) }={max(h(V), ... ,h(V »}n 1 n 1 n
== {h (max (V , ... , V )},1 n
where (V) IX) is a sequence of i. i .d. r. v.' s with d. f. FV
' Bym 1
definition 2 it suffices to prove that for k:;tO c{Z } (k)~O (n~).
n
Using the notation introduced in section 1 we obtain
2nikZ 2nikh(max(V, ... ,V» JIX)n 1 n nc {Z } (k) =lEe =lEe = ~k (h (v» dFV(v)
n 0
IX)
J -s 1 -s n-1~k(h(s+logn»e (l-ne) ds
-logn
IX)J~k(h(s+logn»go(s)ds + ~(n),-logn
where
~(n)= IX) ( )-s 1 -s n-1J~k(h(s+logn» e (l-ne) -go(s) ds.
-logn
By dominated convergence we have for kel
Hence
(n~) •
c{Z } (k)=n
IX)J~k(h(s+logn»go(s)dS-logn
+ 0(1) (n~) •
c
In the following lemmas we give sufficient conditions on the
functions h for (3) to hold. These conditions are more easily
6
verifiable. A more general version of lemma 1 is given in Brands
(1991) •
Lemma 1. Let hen. If
(4)
then
(5)
Ico -s
t~ ~k(h(s+t»e dso
colim I ~k(h(s+t»g (s)dst~ 0
-t
o
o
(k;l:O),
holds, and hence so does (3).
Proof: We define for x~O
Jco -s~(x)= ~k(h(s»e ds
xand W(x)=sup{ /eYt(y) I :yi!:x}.
Substitution of s=y-t in (4) yields
co coJ~k(h(s+t»e-sdS = etI ~k(h(y»e-YdYo t
Let xeR+. Then in (5) we have
te ~(t) •
x-t
I I ~k(h(S+t»go(S)dSI-t
Integrating by parts we find
G (x).t
I Ico~k(h(S+t»go(S)dSI = I Ico~k(h(S»gt (s)dslx
x-t
~ W(x) (gt (x) + ICOlg~ (S)+gt (s) IdS)x
and so it follows that
7
~(x) (l-G (x»,t
00II ~k(h(S»gt (S)dsl ~o
G (x) + W(x) (l-G (x».t t
1 1/2tTaking x=-t, and writing e(t)=exp(-e ) we get
2
IIoo~k(h(S+t»gO(S)dSI ~-t
(t~) •
c
c
This lemma can be interpreted as follows: if h (V+t)~, then
h«Y+t)+)~ (t~), where Y has d.f. Go.
dCoro~~ary. Let X = h(V) for some heH, and V~Exp(l). If condition
(4) holds, then {Z }~U (n~).n
Lemma 2. Let heH. If
(6)
exists, then condition (4) holds, and hence so does (3).
00
Proof: Let £>0. Since I ~k(h(s»ds~O (t~), there is a constantt
A>O such that
00II ~k(h(s»dsl<£t
(t~A) .
Integrating by parts we have
whence for t~
8
[]
We note that for1 2 2 2
h (x) =-x --eosx4 x
(x~8) condition (6) fails,
whereas (4) holds (cf. Brands (1991».
Lemma 3. Let h:R+~ be a continuous non-decreasing function with
inverse function h, and suppose h"(x) exists if x~A, for some AeR.
If
(7)
and
lim h' (x)=O,x~
co
(8) J Ih" (x) Idx < co,A
then condition (3) holds.
Proof: By substituting s=h(x) and setting c=h(O) we have for k~O
Integrating by parts we find
which exists because of the conditions (7) and (8). Now lemma 2
yields the assertion.[]
Next, we state a corollary of theorem 1 and lemma 3, which says
that (z ) is a.u.d. (mod 1) for distributions with hazard ratesn
that decrease monotonically to zero (i.e. h' (x)lO as x~) .
9
Theorem 2. Let X ~ h(V) with positive derivative F', and V-Exp(l).
Suppose h"(x) exists if x~A, for some AeR. If conditions (7) and
(8) hold, then {Z }~ (n~).n
4. Bxamp~es and fina~ remarks.
c
We give some explicit examples to illustrate the scope of the
results of section 3.
1. For the Pareto d.t. F(x)=l-x-(3, x>l, (3)0, (2) holds with
x/(3 - -h(x)=e , and h(x)=(3logx, h' (x)=(3/x. Clearly h(x) satisfies the
assumptions of theorem 2, and thus {Z }~U (n~).n
v - v2. For F(x)=l-exp(-x ), x>O, O<v<l, we have h(x)=x , and theorem 2
dyields {Zn}~ (n~). If v=l, then for F=F
Vit is shown in Jagers
(1990) that {Z } diverges. If v>l, it is also possible to shown
- V-Ithat {Z } diverges. Also, if v~l, we see that h' (x)=vx does not
n
converge to zero as x~, and that condition (8) does not hold. So,
it would seem that conditions (7), (8), and hence condition (4)
are not too far away from being necessary for (Z ) to be a.u.d.n
(mod 1) .
1 1/«3. For F(x)=l---
l---, x>e , «>0, the conditions of theorem 2 are« ogx
satisfied. Thus (Z) is a.u.d. (mod 1) even though F does notn
belong to the domain of attraction of an extreme value
distribution.
10
Obviously, example 2 suggests that there is a relation with
extreme value theory. In fact, if F is absolutely continuous and
belongs to the domain of attraction of G, and if 1/h' (x) tendso
to some c~O monotonically as x~, then {Z} diverges inn
construct a sequence of i.i.d. r.v.'s
distribution. These results, which need rather long proofs, will
be published elsewhere.
Finally, for any r.v. W with P(OSW<l)=l it is possible to
(X ) Ill) such that {Z }~mIn
(n~) (see Brands, Steutel and Wilms (1992».
Acknowledgement.
The authors are indebted to F.W. Steutel and J.G.F. Thiemann for
suggesting the problem treated here, and for reading an earlier
version of this paper.
11
References.
BRANDS, J.J.A.M. (1991), An asymptotic problem in extremal
processes, RANA 91-10, University of Technology, Eindhoven,
The Netherlands.
BRANDS, J.J .A.M., F .W. STEUTEL and R.J .G. WILMS (1992), On the
number of maxima in a discrete sample, Memorandum COSOR
92-16, University of Technology, Eindhoven, The Netherlands
(submitted for publication) .
HOLEWIJN, P.J. (1969), Note on Weyl's criterion and the uniform
distribution of independent random variables, The Annals of
Mathematical Statistics, 40, 3, 1124-1125.
JAGERS, A.A., (1990), Solution of problem 247, Statistica
Neerlandica, 44, 180.
KUIPERS, L. and H. NIEDERREITER (1974), Uniform distribution of
sequences, Wiley, New York.
RESNICK, S.I. (1987), Extreme values, regular variation and point
processes, Springer-Verlag, New York.
RIPLEY, B.D. (1987), Stochastic Simulation, Wiley, New York.
SCHATTE, P. (1983), On sums modulo 21t of independent random
variables, Mathematische Nachrichten, 110, 243-262.
ZYGMUND, A. (1968), Trigonometric Series, Vol. I & II, Cambridge
University Press, New york.
12
EINDHOVEN UNIVERSITY OF TECHNOLOGYDepartment of Mathematics and Computing SciencePROBABILITY THEORY, STATISTICS, OPERATIONS RESEARCHAND SYSTEMS THEORYP.O. Box 5135600 MB Eindhoven, The Netherlands
Secretariate: Dommelbuilding 0.03Telephone : 040-473130
-List of CaSaR-memoranda - 1991
Number
91-01
91-02
91-03
91-04
91-05
91-06
91-07
91-08
91-09
91-10
91-11
January
January
January
January
February
March
March
April
May
May
May
Author
M.W.I. van KraaijW.Z. VenemaJ. Wessels
M.W.I. van KraaijW.Z. VenemaJ. Wessels
M.W.P. Savelsbergh
M.W.I. van Kraaij
G.L. NemhauserM.W.P. Savelsbergh
R.J.G. Wilms
F. CoolenR. DekkerA. Smit
P. J. ZwieteringE.H.L. AartsJ. Wessels
P.J. ZwieteringE.H.L. AartsJ. Wessels
P. J. ZwieteringE.H.L. AartsJ. Wessels
F. Coolen
The construction of astrategy for manpowerplanning problems.
Support for problem formulation and evaluation inmanpower planning problems.
The vehicle routing problemwith time windows: minimizing route duration.
Some considerationsconcerning the probleminterpreter of the newmanpower planning systemformasy.
A cutting plane algorithmfor the single machinescheduling problem withrelease times.
Properties of FourierStieltjes sequences ofdistribution with supportin [0, 1) .
Analysis of a two-phaseinspection model withcompeting risks.
The Design and Complexityof Exact Multi-LayeredPerceptrons.
The Classification Capabilities of ExactTwo-Layered Peceptrons.
Sorting With A Neural Net.
On some misconceptionsabout subjective probability and Bayesian inference.
COSOR-MEMORANDA (2)
91-12
91-13
91-14
91-15
91-16
91-17
91-18
91-19
91-20
91-21
91-22
91-23
May
May
June
July
July
August
August
August
September
September
September
September
P. van der Laan
I.J.B.F. AdanG.J. van HoutumJ. WesselsW.H.M. Zijm
J. KorstE. AartsJ.K. LenstraJ. Wessels
P.J. ZwieteringM.J.A.L. van KraaijE.H.L. AartsJ. Wessels
P. DeheuvelsJ.H.J. Einmahl
M.W.P. SavelsberghG.C. SigismondiG.L. Nemhauser
M.W.P. SavelsberghG.C. SigismondiG.L. Nemhauser
P. van der Laan
P. van der Laan
E. LevnerA.S. Nemirovsky
R.J.M. VaessensE.H.L. AartsJ.H. van Lint
P. van der Laan
Two-stage selectionprocedures with attentionto screening.
A compensation procedurefor mu1tiprogrammdngqueues.
Periodic assignment andgraph colouring.
Neural Networks andProduction Planning.
Approximations and TwoSample Tests Based onp - P and Q - Q Plots ofthe Kaplan-Meier Estimators of Lifetime Distributions.
Functional description ofMINTO, a Mixed INTegerOptimizer.
MINTO, a Mixed INTegerOptimizer.
The efficiency of subsetselection of an almostbest treatment.
Subset selection for an-best population:
efficiency results.
A network flow algorithmfor just-in-time projectscheduling.
Genetic Algorithms inCoding Theory - A Tablefor A) (n, d) .
Distribution theory forselection from logisticpopUlations.
COSOR-MEMORANDA (3)
91-24
91-25
91-26
91-27
91-28
91-29
October
October
October
October
October
November
I.J.B.F. AdanJ. WesselsW.H.M. Zijm
1. J. B. F. AdanJ. WesselsW.H.M. Zijm
E.E.M. van BerkumP.M. Upperman
R.P. GillesP.H.M. RuysS. Jilin
I.J.B.F. AdanJ. WesselsW.H.M. Zijm
J. Wessels
Matrix-geometric analysisof the shortest queueproblem with thresholdjockeying.
Analysing MultiprogrammingQueues by GeneratingFunctions.
D-optimal designs for anincomplete quadratic model.
Quasi-Networks in SocialRelational Systems.
A Compensation Approach forTwo-dimensional MarkovProcesses.
Tools for the InterfacingBetween Dynamical Problemsand Models withing DecisionSupport Systems.
91-30 November G.L. NemhauserM.W.P. SavelsberghG.C. Sigismondi
91-31 November J.Th.M. Wijnen
91-32 November F.P.A. Coolen
91-33 December S. van HoeselA. Wagelmans
91-34 December S. van HoeselA. Wagelmans
Constraint Classificationfor Mixed Integer Programming Formulations.
Taguchi Methods.
The Theory of ImpreciseProbabilities: Some Resultsfor Distribution Functions,Densities, Hazard Rates andHazard Functions.
On the P-coverage problemon the real line.
On setup cost reduction inthe economic lot-sizingmodel without speculativemotives.
91-35
91-36
December
December
S. van HoeselA. Wagelmans
F.P.A. Coolen
On the complexity of postoptimality analysis of0/1 programs.
-Imprecise Conjugate PriorDensities for the OneParameter ExponentialFamily of Distributions.
COSOR-MEMORANDA (4)
91-37
91-38
December
December
J. Wessels
J.J.A.M. BrandsR.J.G. Wilms
Decision systems; therelation between problemspecification and mathematical analysis.
On the asymptoticallyuniform distribution modulo1 of extreme order statistics. (revised version) •