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.

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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

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January

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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

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September

September

September

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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) •