Records: Mathematical Theory (Translations of Mathematical Monographs)

Translations of

MATHEMATICAL MONOGRAPHS Volinuc 194

Records: Mathematical Theory Valero B. Nevzorov

AMCHcan malileIUULIcai JOCICLy

Selected Titles in This Series 194 Valery B. Nevzorov, Records: Mathematical theory, 2001 193 Toshio Nishino, Function theory in several complex variables, 2001 192 Yu. P. Solovyov and E. V. Troitsky, C'-algebras and elliptic operators in differential topology, 2001

191 Shun-ichi Amari and Hiroshi Nagaoka, Methods of information geometry, 2000 190 Alexander N. Starkov, Dynamical systems on homogeneous spaces, 2000 189 Mitsuru Ikawa, Hyperbolic partial differential equations and wave phenomena, 2000 188 V. V. Buldygin and Yu. V. Kozachenko, Metric characterization of random variables and random processes, 2000

187 A. V. Fursikov, Optimal control of distributed systems. Theory and applications, 2000

186 Kazuya Kato, Nobushige Kurokawa, and Takeshi Saito, Number theory 1: Fermat's dream, 2000 185 Kenji Ueno, Algebraic Geometry 1: From algebraic varieties to schemes, 1999

184 A. V. Mel'nikov, Financial markets, 1999 183 Hajime Sato, Algebraic topology: an intuitive approach, 1999 182 I. S. Krasil'shchik and A. M. Vinogradov, Editors, Symmetries and conservation laws for differential equations of mathematical physics, 1999

181 Ya. G. Berkovich and E. M. Zhmud', Characters of finite groups. Part 2, 1999 180 A. A. Milyutin and N. P. Osmolovskii, Calculus of variations and optimal control, 1998

179 V. E. VoskresenskiT, Algebraic groups and their birational invariants, 1998 178 Mitsuo Morimoto, Analytic functionals on the sphere, 1998 177 Satoru Igari, Real analysis-with an introduction to wavelet theory, 1998 176 L. M. Lerman and Ya. L. Umanskiy, Four-dimensional integrable Hamiltonian systems with simple singular points (topological aspects), 1998 175 S. K. Godunov, Modern aspects of linear algebra, 1998 174 Ya-Zhe Chen and Lan-Cheng Wu, Second order elliptic equations and elliptic systems, 1998

173 Yu. A. Davydov, M. A. Lifshits, and N. V. Smorodina, Local properties of distributions of stochastic functionals, 1998

172 Ya. G. Berkovich and E. M. Zhmud', Characters of finite groups. Part 1, 1998 171 E. M. Landis, Second order equations of elliptic and parabolic type, 1998 170 Viktor Prasolov and Yuri Solovyev, Elliptic functions and elliptic integrals, 1997 169 S. K. Godunov, Ordinary differential equations with constant coefficient, 1997 168 Junjiro Noguchi, Introduction to complex analysis, 1998 167 Masaya Yamaguti, Masayoshi Hata, and Jun Kigami, Mathematics of fractals, 1997 166 Kenji Ueno, An introduction to algebraic geometry, 1997 165 V. V. Ishkhanov, B. B. Lur'e, and D. K. Faddeev, The embedding problem in Galois theory, 1997

164 E. I. Gordon, Nonstandard methods in commutative harmonic analysis, 1997

163 A. Ya. Dorogovtsev, D. S. Silvestrov, A. V. Skorokhod, and M. I. Yadrenko, Probability theory: Collection of problems, 1997

162 M. V. Boldin, G. I. Simonova, and Yu. N. Tyurin, Sign-based methods in linear statistical models, 1997 161 Michael Blank, Discreteness and continuity in problems of chaotic dynamics, 1997 160 V. G. OsmolovskiT, Linear and nonlinear perturbations of the operator div, 1997 159 S. Ya. Khavinson, Best approximation by linear superpositions (approximate nomography), 1997

(Continued in the back of this publication)

Translations of

MATHEMATICAL MONOGRAPHS Volume 194

Records: Mathematical Theory Valery B. Nevzorov

R American Mathematical Society Providence, Rhode Island

EDITORIAL COMMITTEE AMS Subcommittee Robert D. MacPherson Grigorii A. Margulis James D. Stasheff (Chair)

ASL Subcommittee Steffen Lempp (Chair) IMS Subcommittee Mark I. Freidlin (Chair) B. B. Hes3opos PEKOPIIbI: MATEMATI4LIECKASI TEOP14A Translated from the Russian manuscript by D. M. Chibisov 2000 Mathematics Subject Classification. Primary 62-01, 62-02; Secondary 62E10, 62G20, 62G30. ABSTRACT. The book presents the theory of records which has been the subject of an intense research activity since 1952. It may be viewed as a research monograph with an extensive bibliography for probabilists and statisticians, actuarial mathematicians, meteorologists, hydrologists, reliability engineers, sports and market analysts. At the same time the book includes a wide variety of exercises and is written in a form which allows it to be used as a textbook for graduate students.

Library of Congress Cataloging-in-Publication Data Nevzorov, Valery B., 1946Records: mathematical theory / Valery B. Nevzorov; [translated from the Russian manuscript by D.M. Chibisov]. p. cm. - (Translations of mathematical monographs, ISSN 0065-9282; v. 194) Includes bibliographical references. ISBN 0-8218-1945-3 (alk. paper) 1. Order statistics. I. Title. II. Series. QA278.7.N48 2000 519.5-dc2l 00-061821

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Assistant to the Publisher, American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permission®ams.org. © 2001 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. Q The paper used in this book is acid-free and falls within the guidelines established to Nnsure permanence and durability.

Visit the AMS home page at URL: http://wv.ams.org/ 10987654321 060504030201

To my parents

Contents Preface

ix

Introduction. Lecture 1

1

Part 1. Order Statistics Lecture 2. Order Statistics and Their Distributions Lecture 3. Three Classical Representations for Order Statistics Lecture 4. Markov Property and Conditional Independence of Order Statistics Lecture 5. Order Statistics for Nonstationary Sequences Lecture 6. Moments of Order Statistics Lecture 7. Moments of Order Statistics (continued) Lecture 8. Asymptotic Distributions of the Middle Order Statistics Lecture 9. Asymptotic Distributions of the Middle Order Statistics (continued) Lecture 10. Asymptotic Distributions of Maxima Lecture 11. Domains of Attraction of the Limiting Distributions of Extremes Lecture 12. Some Topics Related to the Distribution of Extremes

Part 2. Record Times and Record Values Lecture 13. Maxima and Records Lecture 14. Distributions of Record Times Lecture 15. Distributions of Record Values Lecture 16. Shorrock's Representation for Discrete Records Lecture 17. Joint Distributions of Record Times and Record Values Lecture 18. Asymptotic Distributions of Record Values Lecture 19. The kth Records Lecture 20. Generating Function of the kth Record Times Lecture 21. Moment Characteristics of the kth Record Times Lecture 22. Tata's Representation and Its Generalizations Lecture 23. Correlation Coefficients for Records Lecture 24. Records for Nonstationary Sequences of Random Variables Lecture 25. Record Times in the Fe-Scheme Lecture 26. Independence of Record Indicators and Maxima Lecture 27. Asymptotic Distribution of Record Values in the Fa-Scheme Lecture 28. Records in Sequences of Dependent Random Variables Lecture 29. Records and the Secretary Problem Lecture 30. Statistical Procedures Related to Records vii

5 5 10 15 18 21 26

32 38

42

47 50 55 55 59 65 69 76 79 82 86 89 92 95 101 107 114 116 119 125 129

viii

CONTENTS

Appendix 1. Theory of Records: Historical Review

133

Appendix 2. Hints, Solutions, and Answers

137

Bibliography

153

Preface In 2002 the mathematical theory of records will mark its 50th anniversary. The first paper on this subject by Chandler (1952) attracted the attention of many researchers and inspired many new publications. It happens sometimes that, by novelty of its subject or brilliant results, a mathematical paper arouses and maintains interest of numerous researchers to a "fashionable" problem for several years or even decades, but as the "vein of gold" is exhausted this interest drops off sharply. This was not the case for records. The number of publications on this subject has been increasing exponentially, doubling about every 10 years (around 80 in 1977, about 160 in 1987, and more than 300 by the beginning of 1998). In the Introduction, we try to explain this phenomenon. Here we only point out that numerous models of records provide a convenient object for application of various mathemat-

ical methods; on the other hand, there is a lot of numerical data on records in sports (national, Olympic, world records), hydrology (for instance, the floods in St. Petersburg have been recorded for 300 years), meteorology (the reader of course has heard on radio or TV something like "the air temperature today in our city was the lowest for the last 95 years"), etc., which motivate the mathematicians to build models compatible with the available record observations and to try to predict the future record values. It is difficult to separate the theory of records from the theory of order statistics. Records are especially closely related to extremal order statistics. A systematic ex-

position of the theory of order statistics and extremes can be found in books by H. A. David "Order Statistics" (1970, 1981) and J. Galambos "The Asymptotic Theory of Extreme Order Statistics" (1978, 1987). Regarding records, a comparatively detailed review of results (without proofs) and related bibliography can be found only in the form of articles (Nevzorov (1987), Nagaraja (1988), Nevzorov and Balakrishnan (1998)). As mentioned above, the number of publications has practically doubled for the last 10 years. Hence it becomes necessary to relate the classical results for records with the latest advances. This book can be viewed as an "Introduction into the Theory of Records." We tried to present the material in the form combining the features of a textbook and a survey of literature. Thus, after learning the basic methods utilized in the theory of records, the reader will find in Appendix 1 bibliographical notes which, together with the list of references comprising about 300 papers on records and related topics, will allow him to acquire a deeper knowledge of the subject. Moreover, this book contains about 50 exercises which will allow the reader to assess the degree of his mastering the material. Hints and solutions are collected in Appendix 2. The book is written on an intermediate level which presumes the knowledge of only standard courses of probability theory and mathematical statistics. In the first part of the book we present briefly the necessary material on order statistics ix

x

PREFACE

which is used in the theory of records. The reader interested in a more detailed knowledge of the theory of order statistics and its applications is referred to the book by David (1970) mentioned above and references therein. The book can be used for preparing courses on order statistics and records, as well as for studying these areas of probability theory and mathematical statistics on one's own. A large part of the material was employed by the author in lecture courses on order statistics and records at the Faculty of Mathematics and Mechanics

at the St. Petersburg State University and the Department of Statistics at Ohio State University. I hope that the methods for analysis of records, numerous record models, and

the various applications of records treated in the book will attract the attention not only of mathematicians, but also of engineers (especially those engaged in reliability of constructions), actuaries, sport statisticians, specialists in hydrology, meteorology, gerontology, and many others dealing with analysis of extremal values of various random variables and processes. Writing the book gives me an opportunity to set out systematically the results of my research on order statistics and records. It is my pleasure to express gratitude to V. V. Petrov from whom I always received attention and support beginning with my first steps in science. I am indebted to my former students A. V. Stepanov and S. V. Malov, now scientists themselves, who have been the most scrupulous readers of my works, and with whom constant communication stimulated my research in this area. Of great importance for my scientific activity was an exchange of ideas with colleagues. I am thankful to M. Ahsanullah, N. Balakrishnan, P. Deheuvels, V. A. Egorov, G. Haiman, and H. N. Nagaraja with whom I obtained a number of joint results included in this book.

Comments useful in improving the text were made by S. M. Ananievskii, D. M. Chibisov, O. V. Rusakov, and V. V. Slavova. Advice from A. M. Trevgoda and assistance from my wife Lyudmila and my son Igor helped to speed up the process of typesetting the manuscript.

V. B. Nevzorov

St. Petersburg, April 2000

Introduction

Lecture 1

One of the most popular books in the world, after the Holy Bible, is the "Guinness Book of Records" first published in September 1955. For almost half a century this book, regularly updated and republished, stimulated the appearance of many similar publications, such as the "St. Petersburg Book of Records" first published in 1995. Why are records so popular? Maybe this is because we often encounter them in

everyday life, single out the records from the multitude of data, fix and memorize the record values. Of particular interest are records in sports and record values related to natural phenomena. The spectators are drawn to stadiums not only by competition excitement, but also by an opportunity to witness new record achievements which broaden the horizons of human abilities. Indeed, for the majority of people who watch sport competitions or participate in them, achievement of a new record is undoubtedly associated with progress. In order to attract potential record-breakers (and, in turn, numerous spectators and advertisers), the organizers of many athletic competitions set special prizes for breaking records, which are, as a rule, of higher value than the prizes for the winners. One can frequently see a show where daredevils risk their necks trying to break a speed record on water, ground, or in the air, or to leap a motorcycle over a record number of cars, or to achieve a record depth without using an aqualung, in order to place their names into the book of records. After Sir Edmund Hillary and Tenzing Norgay reached the summit of Mt. Everest in 1953, having thus achieved the absolute record in mountaineering for our planet, other mountaineers found the "way out" in climbing Everest or other mountains along the routes of record difficulty. People are excited by records related to geography and the various natural or social phenomena. Tourists strive to see the highest waterfall in the world, the oldest tree, or the largest cave. Cities compete with one another by erecting the highest skyscraper or building the largest stadium. Journalists will not miss a chance to report on the birthday of the oldest person in the world, to describe the life of the richest person, or to show the consequences of the most devastating hurricane in a particular region. In weather reports, newspapers publish the lowest and highest temperatures observed in the locality for the last, say, 100 years, and readers are interested to see if these local records would be broken, while old-timers tell stories to envying listeners about the extremely cold winter they survived some sixty years ago. In English the word "record" means not only the unsurpassed performance in some area, but also a report, account, chronicle, diary, or relics of the past. One can see that all these meanings are logically interrelated. I

I. INTRODUCTION

2

Each record "achievement" - such as the oldest record in the Guinness Book of Records which is due to St. Simeon Stylites the Younger who lived for 45 years standing on top of a stone pillar near Antioch, Syria, some 14 centuries ago; the record miracle near the Sea of Galilee when five thousand men, without counting women and children, were fed with five loaves of bread and two fishes; the pandemic

of plague in Europe in XIV century that claimed about 75 million lives; or the fantastic Bob Beamon's long jump of 8.90 m in 1968 - is registered in chronicles and accounts or is preserved in people's memories. Even in mathematics, record achievements are not infrequent. It is well known, for instance, that the sequence of prime numbers is infinite and there is no largest prime number. Hence, attempts to demonstrate the largest particular prime number continue persistently. The enthusiasm of researchers combined with computer equipment progress resulted in obtaining successively in 1996-1999 the following record values of prime numbers: 21257787 _

1,

21398269

-

1,

22976221

-1,

23021377 _ 1, and 26972593 - 1.

The last of them, found by N. Hajratwall, G. Woltman, S. Kurovski, et at. on June 1, 1999, consists of 2,098,960 digits for decimal representation. Maybe by now the reader can write down a much larger prime number.

Another example refers to the probability theory. A longstanding research was devoted to evaluation of the absolute constant in the well-known Berry-Esseen inequality which sets a bound on the difference between the normal distribution and the distribution of a sum of independent identically distributed random variables

with finite third moment. Esseen (1956) has shown that this constant is no less than

3+ 6

10 27r

The attempts to approach this value were made by Berry, Esseen, Bergstrom, Zolotarev, and others. Making substantial use of computer facilities, Shiganov (1982) obtained for this constant the value 0.7655 lowering by 0.032 the previous record 0.7975 which was due to van Beek (1972). Thus it took 10 years to beat the record. The records as such are memorials of their time. The annals of records reflect

the progress in science and technology and enable us to study and forecast the evolution of mankind on the basis of record achievements in various areas of its activity.

This motivates the necessity to construct mathematical models of records and to develop the corresponding mathematical theory. In 1952 there appeared the first paper on the mathematical theory of records by Chandler (1952). Since then there were many mathematicians who devoted their attention to this subject. All of them, working in different areas of probability theory and mathematical statistics, found interesting problems related to records which required diverse mathematical methods. Presently the author's card-file contains more than 300 papers, half of which appeared during the last 10 years. In a number of them, mathematical models for the analysis of growth of Olympic or world records were suggested (with special attention given to the mile run) and, based on this analysis, attempts were made to predict future records. While the forecast of sports achievements serves mostly for comprehension and is important primarily for the authors of record models themselves as a possibility to assess the fit of the models to real processes, a

I. INTRODUCTION

forecast of the scale of the next record earthquake or flood would make it possible to take adequate precautions. Statistics of the various records contains a large number of data which sometimes cover a very long time interval. Hence a statistician processing the data often deals with a censored sample containing only the record values. This happens, for example, in estimating the durability of some devices in the situation where the production process is modified after each failure. A similar situation arises in processing sports data when out of the multitude of data registered in numerous competitions of different level, only the most valuable and informative ones are preserved for history, and, of course, records are this kind of data. It turns out, surprisingly, that the theory of records is connected with cycles of random permutations, with some algorithms of linear programming, with forming platoons (caravans) of vehicles (when moving in a single lane road for a long time, the vehicles form platoons whose leaders' speeds are a sequence of lower record values), with the optimal selection problem (the so-called secretary problem).

The theory of records relies largely on the theory of order statistics, and is especially closely connected to extreme order statistics. Hence we will discuss first the order statistics and their properties.

PART 1

Order Statistics Lecture 2. ORDER STATISTICS AND THEIR DISTRIBUTIONS

Consider n random variables X1,. .. , Xn. Without loss of generality, we will assume that all of them are defined on the same probability space (f2 = {w}, _F, P). Having arranged these variables in increasing order, we obtain order statistics

Xln 0 and 0 > 0, so that we have found actually the right limit P XI + 0, x2 + 0). But taking into account continuity of f at the points x1 and x2 we can obtain in a similar way the same expression for the limits f (XI + 0,X2 - 0), A XI - 0,X2 + 0), and f (x1 - 0,x2 - 0).

EXAMPLE 2.1. Let F(x) = x, 0 < x < 1, which corresponds to the uniform distribution on the interval [0, 1]. Then (2.10)

fk:n(X) =

k)!xk-1(1

(k - 1)!(n -

-

0 < x < 1,

x)n-k,

i.e., the order statistic Uk,n related to i.i.d. uniform random variables U1, U2,..., Un

has the beta distribution with parameters k and n - k + 1, and (2.11)

f1,2,...,n:n (X1, ... , xn) = n!,

0 < x1 < ... < xn < 1.

EXERCISE 2.3. Let independent random variables XI, X2, ... , Xn have geometric distributions with parameters P1, p2, . , pn (0 < pi < 1, i = 1, 2, ... , n), i.e.,

P{Xi=m}=(1-pa)pi"

m=0,1,....

Find the distribution of the random variable Y, = min{XI, X2, ... , X,, I.

EXERCISE 2.4. Find the joint distribution of the maxima 1lln = max{XI,... , X,,} and Mn+i = max{XI,... , Xn+1 }, where XI, X2, ... are i.i.d. random variables with d.f. F. EXERCISE 2.5. Let i.i.d. random variables X1, X2,. distribution, i.e.,

F(-x) = 1 - F(x + 0),

.., X,, have a symmetric

x > 0.

Show that in this case Xk,n and -Xn_k+l,n are equally distributed and that the sample median Xn+1,2n+1 also has a symmetric distribution.

I.

H)

I)I)DIili s'I'AI'IS'rIC'S

Lecture 3. THREE CLASSICAL REPRESENTATIONS FOR ORDER STATISTICS

We pointed out in Lecture 2 that the convenient independence property is lost when we pass from original independent random variables X1, X...... X to order statistics X1,n < ... < X,,n. Nevertheless, in the particular cases of exponential and uniform distributions, the order statistics can be expressed in terms of sums of independent random variables. One more important result consists of the possibility to pass from order statistics related to some particular distribution (e.g., uniform) to arbitrary order statistics by means of monotone transformations. REPRESENTATION 3.1. Let X j, 5 ... 0.

The above relations are particularly important for the reliability theory. Let a system consist of n elements of the same kind working simultaneously, whose life times are mutually independent and have the same exponential distribution with parameter A. Then (3.6) implies that the time intervals between consecutive failures of the elements are independent and have exponential distributions (with corresponding parameters A/n, A/(n - 1), ... , A/2, A). PROOF OF (3.4). It suffices to check that the vectors we compare have the same density function. Putting f (x) = exp(-x), x > 0 into (2.9) we obtain that the joint density of order statistics Zl,n, Z2,,

,

, Zn.n has the form n

(3.8)

11.2....... :n(xl, x2, ... , xn) = n! fl exp(-xy) = n! exp ( - L. X

\ 1

s=1

x=1

if 0 < X1 < x2 < < x,, < oo. The joint density of i.i.d. exponential random variables v1, v2, ... , v, in the domain A = {yy > 0, s = 1, 2, .... n} is given by (3.9)

y,).

9(yl,y2,...,yn) = exp s=1

I. ORDER. STATISTICS

12

The linear change of variables

(v1,uz,...,vn) = ( yt

J2

y1

y2

L,

.. +-yn)

n' n +n-1' n +n-1

with Jacobian 1/n!, which corresponds to the passage to random variables

V1 = n,V2= n

+n121,...,Vn=

+nv21+...+un,

n

has the property

Vl +v2+...+Vn

=y1+y2+...+Yn

< vn < oc. Now (3.9)

and maps the domain A into the domain 0 < v1 < v2 < implies that V1, V2, ... , Vn have the joint density n

0 cm.Urn,n. m=1

1. J.\RKOV VROPEIRTY AND CONDITIONAL INDEPENDENCE

15

Lecture 4. MARKOV PROPERTY AND CONDITIONAL INDEPENDENCE OF ORDER STATISTICS

As we pointed out in Lecture 2, when we pass from the original random variables

X1, X2, ... to order statistics, the independence property is lost. It can be shown, however, that for continuous underlying distributions the order statistics form a Markov chain.

We will restrict the proof of the Markov property of order statistics to the case of absolutely continuous underlying distributions. Let the random variables X1, ... , Xn have a common density f. In this case, in order to prove that Xl n, ... , Xn_n form a Markov chain it suffices to show that for any 1 < k < n the conditional density (to be denoted by x1, ... , xk)) of Xk+I,n given all previous order statistics, X1.n = x1, ... , f (U Xk,n = xk, coincides with conditional density f (u I Xk) of Xk+l,n given only that Xk,n = xk Using notation introduced in Lecture 2 we can write the conditional densities f(u I x1,...,xk) and /f (u I xk) as I

(4.1)

flu I X1,...,xk) =

f1,...,k+1:n(x1,...,xk,u) f1....,k:n(x1, ... , xk)

and

(4.2)

f(u I Xk) =

fk,k+l:n(xk,u) fk:n(Xk)

It remains to employ formula (2.8) and check that the right-hand sides of (4.1) and (4.2) are equal to F(u))n-k-l

(n - k)(1 -

(1 - F(xk ))

n-k

f(u) ,

10,

k=0,1,2.

To simplify the proof we will also assume that P{X < 0} = 0. One can easily verify the following equalities:

P{X3.3=2,X2.3=1,X1,3=1}=3pip2, P{X2.3 = 1, X1,3 = 1} = Pi + 3p2 i (1 - pn - PI), P{X3,3 = 2, X2,3 = 1} = 3P1P2 + 6poPIP2, and P{X2.3 = 1} =P3 + 3Pi(1

-

P()

- pl) + 3PoP1 + 6Pop1(1 - po - pl)

1. ORDER sTA'rls'rucs

16

This implies that

P{X3.3=21 X2,:s=1,X1,3=1}=

P{X3,3 = 2, X2.3 = 1, X1.:1 = 1 }

P{X2,3=1,X1.3=1} 3p2 pi + 3(1 - po - p1)

and similarly 3p1p2 + 6poP2

P{X3,3 = 2 I X2,3 = 1} =

Pi + 3p1(1 - po - pi) + 3pop1 + 6po(1 - po - pi)

Let us show that P{X3,3 = 2 1 X2.3 = 1,X1,3 = 1} > P{X3.3 = 2 I X2.3 = 1}.

For that it suffices to establish that

(p1+3(1-po-pl))(pl+2po) < (pi+3p1(1-po-pl)+3pop1+6po(1-po-pl)), but this relation is equivalent to the obvious inequality P, (p, + 2po) < pl + 3pop1,

which completes the proof. The following theorem plays an important role in the treatment of order statistics and especially their sums (which is often needed when dealing with truncated samples arising under the various censorship models).

THEOREM 4.1. Let X1,,, < ... < be order statistics corresponding to a continuous d.f. F. Then for any 1 < k < n the random vectors

X") = (X1,n,...,Xk-l,n)

and

X(2)

= (Xk+l,n, ... , Xn.n)

are conditionally independent given any fixed value of the order statistic Xk,n. Fur-

thermore, the conditional distribution of the vector XM given that Xk,n = u coincides with the unconditional distribution of order statistics Y1.k_I, ...,Yk-l.k-1 corresponding to i. i. d. random variables Y1,. .. , Yk_ 1 with common F(u) (x)

=

F(u),

d. f.

x < U.

Similarly, the conditional distribution of the vector X(2) given the same condition coincides with unconditional distribution of order statistics W1.n-k, ... , Wn-k.n-k related to the d. f. F(x) - F(u) F. (x) =

1 - F(u)

x > U.

4. M\R.KOV PItOPEIUY AND CONDITIONAL. INDEPENDENCE

17

PROOF. To simplify the proof, we will assume additionally that the underlying random variables X 1 .... , XU have a density function f. We see from (2.6) and (2.9) that the conditional density function of order statistics X1. ... , Xk_ 1 , Xk+l, ... , Xn given that Xk = u (to be denoted by f (xl, , xk_1, xk+1, - . xn I u)) has the form - -

f (x

1

,

-

,

xk -

1

,

x 1+

I

,

-,

xn I u) =

f1,2,...,n:n (xl, x2, - .. , xk-1, xk+1,.. , xn ) fk:n(u) n k-1

_ (k - 1)!J1

(4.3)

f(xs)(n

s=1

F(u)

- F(

- k)! II

F(u r=k+1) 1

Since f (x)/F(u), x < u, and f (x)/(1 - F(u)), x > u, are density functions corresponding to d.f.'s Fu and F(u) respectively, all assertions of Theorem 4.1 follow from (4.3) and (2.9). 0

The following result similar to Theorem 4.1 can be useful when dealing with two-sided censoring.

THEOREM 4.2. Let Xl,n _< _< Xn,,i be the order statistics related to a continuous d. f. F. Then for any 1 < r < s < n the conditional distribution of order statistics Xr+l,n, , X,-l,n given that Xr,n = y and X3,n = z, y < z, coincides with the unconditional distribution of order statistics Vl,,_r_1i...,V,_r_1,,_r_l corresponding to i.i.d. random variables V1i...,V,_r_1 with common d.f. -

- -

V.,

=(x) = F(x) - F(y)

y < x < z.

F(z) - F(y)'

Theorem 4.2 enables us, for example, to write down the d.f. of

T = Xr+l,n + ... + X,_1.n in the form

P{T < x} = ff

P{T < x I Xr,n = y, Xs.n = z}frs:n(y, z) dydz

o0 0.

Then, as we know, for any n = 1, 2.... relation (3.4) holds:

,- +

Vl

d (Z1,n,Z2,n,...,Zn,n)= n

Vl

V2

n

n -1

V2 ,..., Vl-+ +...+Vnll/ n n 1

where vl, v2,... are i.i.d. exponential random variables with V.

H(x) = max{0,1 - exp(-x)}. Since EVk = 1 and Var vk = 1, k = 1, 2, ... , (3.4) implies that (6.5)

(6.6)

µk:. = E(Zk,n) = E (VI n + n 1

1

n

n-1

n-k+1 1

Var(Zk,n) = Var (I!1 + n

1

vk

v2

n-k+1' 12

n-1

1

n2 + (n --1)2

+ ... +

+ ... +

VR

n-k+1 I 1

(n

- k + 1)2'

and

(6.7)

cov(Zr.n, Zs,n) = E(Zr,n - EZr,n)(Zs,n - EZs,n) = Var(Xr.n) _ 1 1 1 +...+

- n2 + (n-1)2

(n-r+1)2'

r<s.

I. ORDER STATISTICS

24

Uniform distribution. Now consider the order statistics Uj.n < n = 1, 2, ... , related to the sequence of i.i.d. random variables U1, U2

F(x)=x,

< U,,,n,

.. with d.f.

0<x no the following asymptotic relations hold:

E(Zn,n)=

n+ n-1+ 1

1

+1=logn+ ry +o(1),

where ry = 0.577215... is the Euler constant, Var(Zf,n) _ 2 + (n 11)2 +...+1 = (7.11) E(Z(Qn)+1.n)

=1 n

1

}

n- 1

+ ... +

2

6

+O\n

n- (anj = - log(1 - a) + O (n1), 1

and (7.12)

where 0 0). The situation becomes more complicated if f (G(a)) = 0. In this case G(a) is a minimum point of the function f and f'(G(a)) = F(2) (G(a)) = 0 if F is twice differentiable in a neighborhood of the point G(a). Then F(Xk'n)

n+ 1 +F'(G(n+ 1Jl (Xk'n +

2F(2)(G(n+1)1(Xk,n

+1F(3)(G(nk

- G(n+ 111 -G(ri+1)/2

(Xk,n-G(ri+1)13

and

Ulan)+1,n

d

F(X(a)) PL- a + F'(G(a)) (X(,,) - G(a))

+

1

(X(0) - G(a))`

1 F(2)

pa))

+ 1 F(3) (G(a)) (X(") - G(a))3 a + 1 F(3) (G(a)) (X(Q) - G(

In this case it is the random variable 1(X(,7)

- (G(a))3F'(3) (G (a)) a(ln a)

Q))3.

I. ORDER

to

which has the limiting standard normal distribution. Now we will prove the following theorem. THEOREM 9.1. Let the underlying random variables X1, X2, ... have a differentiable density function f such that

sup If'(x)I < M < oo.

where the supremum is taken over the set A = {x: f (x) > 0}. Then for any k = 1, ... , n and n = 1, 2.... the following inequality holds:

sup x

Pl (Xk,n - G(k/(n + 1))) f (G(k/(n + 1)))


0, the following bound holds for the sample quantile X(,,) = X[af]+I.n:

P{ (X(,) -

G(a)) f

(G(a))nl12


1, o>0.

Then

P{Mn/7tt/a < x} =

(1 -

(xnl/°)-°)n

x>0, x 0.

EXAMPLE 10.4. Let

F(x)=1-(1-x)Q,

0<x0

(in particular, for a = 1 this is the uniform distribution on [0,1]). Then

P{(Mn - 1) n l/Ot < x } = ( xn -1/° + 1 ) " = (1

-

_an )

)

exp

(- (- x )°)

if x < 0 and

P{(MM - 1)nl/° < x} = 1 The limiting d.f. in this case has the form

for

x > I.

exp (- (-x)a)

for

x < 0,

1

for

x > 0.

(10.3)

'PQ

(x)

In Examples 10.1-10.4 we obtained three types of possible limiting distributions for maxima. Of course, for any d.f. F one can find a sequence of normalizing

constants {an} such that the sequence of random variables llln/an will have a degenerate limiting distribution. Therefore we are interested in finding all possible nondegenerate limiting distributions for centered and normalized maxima Al,,. They are described in the following theorem by Gnedenko, which summarized the numerous studies in this area. Before formulating it, let us point out that the type of the distribution F, along with the d.f. F itself, comprises all d.f.'s F(ax+b), where a > 0 and b are arbitrary

constants. Two d.f.'s Fl and F2 belong to the same type if there are constants a > 0 and b such that F2 (x) = Fi(ax + b).

1. ORDER STATISTIC'S

44

THEOREM 10.5. The set of all nondegenerate distributions for suitably centered and normalized maxima consists only of distributions that belong to the types A(x), 4),,(x), and `yn(x), a > 0.

REMARK 10.6. For different positive values of the shape parameter a the d.f.'s (%, (and, similarly,',) determine different types of distributions. For the proof of the theorem we need several lemmas. The following lemma is due to Khinchine. LEMMA 10.7. Let a sequence of d.f 's Fn converge weakly to a nondegenerate d. f. G, i.e.,

n - oc, Fn(x) - G(x), for any continuity point of G. So that the sequence of d.f.'s Hn(x) = Fn(anx+bn) converges, for some constants an > 0 and bn, to a nondegenerate d.f. H it is necessary and sufficient that an --+ a, bn --# b as n -+ oc, where a > 0 and b are some constants, and H(x) = G(ax + b). PROOF. Let us prove necessity. Since H is a nondegenerate d.f., one can find

its continuity points y and z (-oo < y < z < oo) such that 0 < H(y) < H(z) < 1. Without loss of generality, we will assume that y = 0 and z = 1 (which can be achieved by a linear transformation). Moreover, one can find continuity points u and v of function G such that

-00 0 and arbitrary e > 0 the inequalities Fn(k) (ax + b - e( x + 1)) < Hn(k) (x) : F,.(k) (ax + b + e(x + 1))

10. ASYMIF''I'O'I'IC DISTRIBUTIONS OF AIAXI%IA

a5

hold for k large enough. Letting n(k) tend to infinity and then e tend to zero, we obtain H(x) = lim Hn(k) (x) = G(ax + b). Similar arguments can be applied for negative x. Thus we have shown that for almost all x (except for at most countably many points)

H(x) = G(ax + b), i.e., H and G belong to the same type of distributions. Now we show that for different sequences n(k), k = 1, 2, ... , tending to infinity the sequences an(k) and bn(k) cannot have limits a1 and b1 different from a and b. Indeed, if there were such a1 and b1, then for almost all x the equality G(ax + b) = G(ai x + b1)

would hold, implying that for any growth point x" of H

ax*+b=alx*+ b1. Since H is nondegenerate, there are at least two such points, hence a1 = a and b1 = b.

The proof of sufficiency is obvious.

We are interested in possible limiting (as n -+ co) d.f.'s H(x) for sequences

Hn(x) = (F(ax + bn))n, where F is the underlying V. and an and bn, n = 1, 2, ... , are some centering and normalizing constants. LEMMA 10.8. In order for a nondegenerate d.f. H to appear as the limit of a sequence (F(anx+bn))n for some d. f. F and constants an > 0 and bn, n = 1, 2, ... , it is necessary and sufficient that for any s > 0 and x (10.4)

H6 (A(s)x + B(s)) = H(x),

where A(s) > 0 and B(s) are some functions defined for s > 0. PROOF. Sufficiency is obvious since if (10.4) holds, one can take F = H as the

underlying V. and an = A(n) and bn = B(n) as the centering and normalizing constants. Let us prove necessity. Of course, along with the relation (10.5)

Hn(x) = (F(anx + bn))n -. H(x),

the relation (10.6)

(F(ain3ix+b(n31))in"l -+H(x),

oc,

n

holds for any s > 0, where [x] denotes the integral part of x. This implies that (10.7)

(F(ain3ix + b(,,3)) n -+ (H(x))1/',

n -+ oo.

Observe that (10.7) can be rewritten as (10.8)

Hn (an(s)x + /3 (s)) -+ (H(x))113,

I. omwit s'l'A'I'IS'I'I('S

16

where

an.(s) = a(,,.I/a > 0 and Nn(s) = b(n.a) - b,,. Comparing (10.5) with (10.8) we obtain by Lemma 10.7 that for any s > 0 there are constants A(s) and B(s) such that an,(s)

A(s) > 0,

fn(s) -+ B(s) > 0,

and

H(x)'I' = H(A(s)x + B(s)).

This completes the proof of Lemma 10.8.

Thus we reduced the problem of finding limiting distributions of maxima to finding all solutions of the functional equation (10.4). This problem can be solved using the following well-known result, which we formulate without proof. The first part of this statement can be found, e.g., in Galambos (1978, 1987), the second part follows easily from the first. LEMMA 10.9. If a monotone function v(t) for all t and s fulfills the equality

v(t + s) = v(t)v(s),

(10.9)

then either v(t) = 0 or v(t) = exp At, where A is a real constant. If v(t) fulfills the equality

v(t + s) = v(t) + v(s),

(10.10)

then v(t) = At, where A is a real constant. Now we can proceed to the proof of Theorem 10.5.

It is seen from Examples 10.1-10.4 that all d.f.'s of types A(x), ' (x), and WQ(x), where a > 0, satisfy (10.4). We will prove that there are no other nondegenerate d.f.'s satisfying (10.4). Taking logarithms of both sides of (10.4) twice, we obtain the equality (10.11)

-log(-log(H(A(s)x+B(s)))) -logs=-log(-logH(x)),

which holds for all x such that 0 < H(x) < 1. Denoting by T(x) the inverse function to V(x) log (- log H(x)) we can rewrite (10.11) in the form

A(s)x + B(s) = T(log s + V(x)). This equality is equivalent to the relation

A(s)T(x) + B(s) = T(log s + x), which implies that

A(s) (T (x) - T(0)) = T(log s + x) - T(log s). Substituting y = logs and denoting AI(y) = A(el)

and

Tj(x) = T(x) - T(0),

we arrive at the equality (10.12)

Ti(x + y) - Tj(y) = A1(y)Ti(x)

Relation (10.12) and the equality Ti(x + y) - Ti(x) = A,(x)TI(y)

1

1. DOMAINS OF ATTRACTION

17

symmetric to (10.12) imply that (10.13)

T1(y)(1 - A, (x)) = 71(x)(1 - A1(y)).

Consider two cases.

(i) Let A1(x) - 1. Then (10.12) implies T1 (x + y) = Ti (x) + Ti (y).

By Lemma 10.9, T! (x) = .\x, which implies that

H(x) = exp{- exp(-,\x + a)},

where \ and a are some constants. Since H(x) is a d.f., we observe that '\ > 0 and H belongs to the same type as A. (ii) Assume now that there is a point x with Al (x) 54 1. We will show that in 1 for any x # 0. Assume that A! (y) = 1 for some y # 0. Then this case A, (x) it follows from (10.13) that Tl (y) = 0, and (10.12) implies that T, (x + y) = T! (x)

for any x, which contradicts the fact that T! (x) has at least one growth point. Consequently, A1(x) 0 1 for all x 34 0. Fix some y = yo

0 and denote

T1(yo) c

1 - A1(yo)'

Then we see from (10.13) that for all x (10.14)

T1(x) = c(1 - A1(x)).

Using (10.14) we deduce from (10.2) the relation (10.15)

A1(x+y) = A1(x)A1(y)

whose solution by Lemma 10.9 is T1(x) = exp Ax. It is obvious that A 0 0. If A > 0, it can be easily shown that H(x) belongs to the same type as the d.f. 4Q(x) for a = 1/A, while for A < 0 the function H(x) belongs to the same type as the d.f. !PQ(x) for a = -1/A. The proof is completed. 0

EXERCISE 10.1. Show that if F(x) = cI (x), then the limiting d.f. for the random variable Mn/n1/Q coincides with (PQ.

EXERCISE 10.2. Show that if F(x) = WQ(x), then the limiting d.f. for the

random variable Mn"' ncoincides with 41Q. Lecture 11. DOMAINS OF ATTRACTION FOR THE LIMITING DISTRIBUTIONS OF EXTREMES

After finding all possible limiting distributions, the second important issue in the theory of extremes is the description of the corresponding domains of attraction D(A), and D(',,) of the three limiting types. That is, for each underlying d.f. F we wish to indicate the corresponding limiting distribution A. 4iQ, or 41Q. It should be pointed out, first of all, that there are d.f.'s F for which the only possible limiting distribution of maxima is the degenerate one. For example, if X1, X2,... are independent and take two values 0 and 1 with probabilities p and 1 - p, then

P{A,IM = 0} = pn and P{Mn = 1} = 1 - pn. Therefore loin can have only the

1. ORDER STATISTICS

48

degenerate limiting distribution at point 1. Less trivial examples of this kind are given by geometric and Poisson distributions. A continuous d.f. which possesses this property is, e.g., F(x) = 1 - I/ log x, x > e. There are necessary and sufficient conditions for an underlying d.f. F to belong to each of the domains of attraction D(A), D(4),,), and D(WY0), but their form is cumbersome and the proofs are tedious (see, e.g., Calambos (1978, 1987)), so that we present only some sufficient conditions and some necessary conditions close to them.

THEOREM 11.1. Let d. f. F have positive derivative for all x > xo. If for some

a>0

lim

(11.1)

xF'(x) = a,

xoo 1 - F(x)

-

then F E D((Da). The centering, bn, and normalizing, an, constants can be taken to be

n=1,2,..., bn=0 and an=G(1- 1), n

(11.2)

where G is the inverse function to F. PROOF. Denote

a(x) =

xF'(x)

1 - F(x)

Since

a(t)

f.x o

t

dt = f x d( - log(1 - F(u)))

log(1- F(x)) + log(1 - F(xo)),

o

we obtain that

1 - F(x) _ (1 - F(xo)) expI( - f

a(t) dt y, t

11

for x > xo, and for an = G(1 - 1/n) we have

1-F(an)=1, n

n=1,2,...,

and

n(1 - F(x)) = exp { - J any a(t) dt1 = exp { - r a(ant) dt y. l t l 1 t Condition (11.1) and the definition of an imply that an -+ oo as n -+ oo and for any fixed t the integrand in the right-hand side of (11.3) tends to a/t, hence n(1 - F(anx)) for any fixed x > 0 tends to x-° as n -+ oo. It remains to observe that

f

(11.3)

(11.4)

Hn(x) = P{ = (

Mn an

< x} _

(F(aux))"

) exp(-x- ) 1 -r-nx-+ n

We have shown that Ha(x) converges to

H(x) = exp(-x-')

as

n-+oc.

It. DOMAINS OF ATTRACTION

49

for any x > 0. Since H(0+) = limxlo H(x) = 0, we have that H,,(x) for x < 0 also tends to zero. Thus we have shown that F belongs to the domain of attraction of the limiting d.f. REMARK 11.2. The following necessary condition which we formulate without proof is close to the sufficient condition in Theorem 11.1: if the d.f. F has a density

which is monotone in the domain {x: 0 < F(x) < 1} and F E D(I ), then (11.1) holds. Note also that if F E D(4),,), then F(x) < 1 for all x < oo. Now we give the corresponding conditions for the limiting d.f. 41Q. THEOREM 11.3. Let d. f. F have positive derivative F'(x) for x in some interval

(x1,xo) and F'(x) = 0 for x > xo. If for some a > 0 we have (11.5)

lim

x-xo

(xo - x)F'(x) = a 1 - F(x)

then F E D(WYa). The centering, bn, and normalizing, an, constants can be taken to be (11.6)

bn=xo

and an=xo-G(1- 1), n

n=1,2,...,

where G is the inverse function to F. The proof is similar to that of Theorem 1(1.1. For x E (x1,lxo) (11.7)

1 - F(x) = (1 - F(xl)) expI( -

where

Jx1

a(t) dt xot

JJJ

a(t) _ (xo - t)F'(t) 1 - F(t)

Since G(1) = xo, we have that an - 0 as n -* oo. It follows from (11.7) that for

x 0. Thus F E D(qla).

I. ORDER STATISTICS

50

REMARK 11.4. It can be shown that if F has a monotone density in the domain {x: 0 < F(x) < 1} and F E D('Y,.r), then condition (11.5) holds, i.e., the sufficient condition given in Theorem 11.3 is close to the necessary and sufficient one. It can

also be shown that if F E D(',,), then sup{x: F(x) < 1} < oc. Now we formulate a theorem on the domain of attraction of A.

THEOREM 11.5. Let d.f. F have negative second derivative F(2)(x) for x in some interval (x1, x11), and let F'(x) = 0 for x > x0. If (11.9)

limo F(2)(x)(1

X

-

F(x))

then F E D(A). The centering, bn, and normalizing, an, constants can be taken to be

(11.10)

bn,=G(1- -') and an=f(bn), n

where

f(t) =

1 - F(t) F'(t)

The proof is similar to the proofs of Theorems 11.1 and 11.3.

REMARK 11.6. It can be shown that if F has a monotone increasing second derivative F(2) (x) and F E D(A), then condition (11.9) holds.

EXERCISE 11.1. Show that if F is the standard normal d.f., then F E D(A), and the centering and normalizing constants can be taken to be bn = (2logn - loglogn - log47r)112

and

an = 1/bn.

EXERCISE 11.2. Show that the Cauchy distribution with d.f.

F(x) = 2 + - arctanx belongs to the domain of attraction D(4?1), and one can take

a=n and b=0. Lecture 12. SOME TOPICS RELATED TO THE DISTRIBUTION OF EXTREMES

In two preceding lectures we studied asymptotic behavior of maximal order statistics. These results can be easily carried over to minima,

Inn = min{X1,...,Xn} = X1_.. This is done by using the following obvious relation between maxima and minima: (12.1)

lnin{X1,...,X} = -max{Y1i...,Y},

Yk = -Xk,

k = 1,2,... .

For positive random variables X1,X2,... the following equality may also be useful: (12.2)

min{X1,... , X,} =

1

max{V1,..., Vn}

,

Vk = 1 . k = 1, 2,... . Xk

12. SOME Rr-'.[.AI'Er) TOPICS

51

It is seen from equality (12.1) that its suitably centered and normalized righthand side can converge in distribution to a limiting d.f. L(x) satisfying the equality L(x) = 1 - H(-x), where H(x) is the limiting d.f. for maxima which corresponds to the underlying d.f.

F(x) = 1 - F(-x + 0) with F(x) = P{X < x}. Accordingly, L(x) must belong to one of three types of limiting distributions for minima, which we will write as (12.3)

(12.4) (12.5)

A(x) = 1 - A(-x) = 1 - exp ( - exp(x)), -oc < x < oc, 1 - exp (- (-x)-a) for x < 0. fa(x) = 1 - 4ia(-x) = S for x > 0, l 1, 1 - exp(-xa) for x > 0,

Wa(x) = 1 - IF,, (-x) =

f

for

0,

x < 0.

where a > 0. _ Denote by D(A), D( $a), and D(Wa) the corresponding domains of attraction of the limiting distributions for minima. It is easily seen that the d.f.'s from the domains of attraction of the limiting laws for maxima and minima are related in the following way: (a)

F E D(A)

F E D(A);

(b)

F E D(4ia)

F E D(4i0);

(c)

F E D(Wa) b F E D('Pa).

Since the asymptotic behavior of maxima and minima is determined by the right and left tails of the underlying d.f. F respectively, one can easily give examples where

_ F E D(A)

and simultaneously F E D(4ia)

or

F E D('Pa) and F E D(Wp) or, say,

F E D(4ia)

but F belongs to none of the domains D(A), D(4ia), D('ia).

Let us point out that if F is the d.f. of a symmetric law, then simultaneously

F E D(H) and F E D(H), where H denotes one of the symbols 4ia, 'Pa, or A. EXERCISE 12.1. Using (12.1) and the material of the previous lecture, find the form of centering and normalizing constants in the limit theorems for minima. EXERCISE 12.2. Show that if

F(x) = 1 - exp(-x),

x > 0,

then F E D(f1). What are centering and normalizing constants in this case? Let us return to the three types, A, 4ia, and %Pa, of limiting distributions for maxima. It turns out that all these distributions can be presented in a unified form

1. ORDER. S 1'A'l'IS'rI( S

52

using the so-called generalized distribution of extreme values. For this it suffices to introduce the one-parameter family of d.f.'s

H(x, a) = exp ( - (1 +

xa)_1"0'),

-00 < a < oo,

which are defined by this formula in the domain 1 + xa > 0 and are equal to zero or one (depending on the sign of a) for 1 + xa < 0. If a > 0, then H(x, a) has

the same type as ' j/Q(x). For a < 0 the d.f. H(x,a) coincides, up to a shift and scale transformation, with W_1/,(x). For a = 0, by H(x,0) is meant the limit of H(x,a) as a -+ 0, and we obtain that H(x,0) is equal to A(x). With all limiting d.f.'s written in this unified form, the problem of determination of the type of limiting distribution based on a sample of maxima reduces to estimation of the unknown parameter a or to testing certain hypotheses about this parameter. After we complete the study of the asymptotic behavior of maxima, let us see what will change if instead of Xn,n or Xl,n we consider the kth maxima Xn-k+1.n or the kth minima Xk,n as n -+ oo with k fixed. We will formulate some results for the sequence of the kth maxima Xn-k+l,n The corresponding results for the kth minima can be immediately obtained from the theorems for Xn_k+l,n if we use the following obvious extension of (12.1):

Xk,n = -Yn-k+l,n, where Yl,n
(- 100. (x))'

(D(k)(x) = pa(x) (12.7)

m!

m=

=0

= exp(-x-a)

k-t -am m=0

M!

'

x>0,

and

(-logT.(x))m

Tak)(x) = Va(x) (12.8)

m!

m=0

= exp(-(-X)a)

(-x) 1am

L m=0

'

x m}On+m = r n = n E k1 = 00.

The form of the right-hand side of (13.1) prompts one more (quite elementary) proof of this equality. Indeed, by continuity of F the independent random variables Xl,... , Xn+m take different values almost sure, and since all of them have the same distribution, the events {Mn+,,,, = Xk}, 1 < k < n + m, are equiprobable, i.e.,

P{ Mn+m =Xk}-n+m 1 It remains to note that

P{ max{Xn+l,... , Xn+m} < Mn} = P{Mn+m = Mn} n

n

= I: P{Mn+m = Xk} = n+m k=1 By symmetry, we will obtain the same distribution as in (13.1) if we consider the additional number of observations needed to get an observation falling into the random interval (-oo,mn), where mn = min{X1....,Xn}. Now, let N(1,n) 55

2. RECORD TIMES AND RECORD VALUES

56

denote the minimal size of the additional sample containing at least one observation falling outside the interval [mn, MnJ. Then

P{N(1,, n) >r m} = P{mn < Xn+k < Mn, k = 1, 2, ... , m} (13.3)

= J x00 Ju (F(v) - F(u))mn(n - 1)(F(v) - F(u))n-2 dF(u) dF(v) = n(n - 1) J

_

0o Ju

F(u))-+n-2

(F(v) -

dF(u) dF(v)

n(n - 1)

(n+m)(n+m- 1) 00

oo

x

I 0o its

(n + m) (n + m - 1) (F(v) - F(u))

n+m-z

dF(u) dF(v)

n(n - 1)

(n+m)(n+m-1) PI-00 < mn+m < Mn+m < oo} _

n(n - 1)

(n+m)(n+m-1) and 00

00

(13.4) E N(1, n) _ > P{N(n) > m} = n(n - 1) m=0

M=0

1 (n+m-1)(n+m)

00

= n(n - 1) E

m=n m(`

1

)

=

n(n - 1) E (m l l

1)=n.

m=n

Like (13.1), the relation (13.3) also admits a simple proof. In this case the problem can be reduced to the following one: out of n + m places, two places are randomly chosen for allocating the maximum and minimum on them, and we have to find the probability that the numbers in both places do not exceed n. An elementary calculation gives for this probability the value

n(n - 1) (n + m)(n + m - 1) It is of interest to compare (13.2) with (13.4). We see that E N(n) = oo already for n = 1, whereas E N(1, n) < oo for any n = 1, 2, .. . Consider the simplest case of the Wilks' problem: we observe the random variable X1; denote L(1) = 1 and find in the sequence X2, X3, ... the first observation (to be denoted by XL(2)) exceeding X1. It follows from (13.1) that

1,2,...,

P{L(2) > (13.5)

P{L(2) = j} = P{L(2) > j - 1} - P{L(2) > j} 1

j = 2,3,...,

and (13.6)

E L(2) = oo.

13. MAXIMA AND RECORDS

57

Along with the random index L(2) define the index

L(2)=min{j> 1: Xj <X1}. Obviously (13.5) and (13.6) remain valid with L(2) replaced by Z(2). It is curious

that P{ min (L(2), L(2)) = 2} = 1 and E (min (L(2), L(2))) = 2, although

E L(2) = E L(2) = oo. The variables L(2) and Z(2) are the upper and lower record times respectively, and

X(2) = XL(2)

and

X(2) = XL(2)

are called the upper and lower record values. In view of (12.1) the results for minima mn and various functionals thereof can be easily obtained (restated) from the corresponding results for maxima Mn. For this reason we will restrict ourselves

to the upper record times and upper record values and refer to them simply as record times and record values. We have already introduced the record time L(2). Define the subsequent record times L(n) by the recursive relation (13.7)

L(n + 1) = min{ j > L(n) : Xi > XL(n) },

n = 2,3,...,

i.e., having observed the sample X1i ... , XL(n) of size L(n) and the maximum ML(n) = XL(n),L(n) = max{X1, ... , XL(n)}, we find in the sequence XL(n)+1, XL(n)+2,

the first random variable (with index L(n + 1)) that exceeds ML(n). The random variables (13.8)

X (n) = XL(n) = XL(n),L(n) = ML(n),

n = 1, 2, ... ,

are called record values. If we replace the strict inequality in (13.7) by ">" we will obtain weak records.

This corresponds to the situation when the repetition of a record mark is also counted as a record. For continuous underlying distributions the passage from ordinary records to weak records does not make any difference. For this reason we will invoke the weak records in Lecture 16 when dealing with record times and record values for discrete distributions. There is another way to define the record times and record values. Consider the nondecreasing sequence

-oo<M1 <M2:5 ... <Mn-1 <Mn n} )112} = P N(no) - log no > -x( n I.

n+xn1/2

(logno)1/2

-+1-(D(-x)=$(x)

as

n

A more detailed derivation using (14.24) enables us to obtain the following inequality: (14.25)

sup IP{IogL(n) - n < xnl/2} - -D(x)I

n'/z'

2. RECORD TIMES AND RECORD VALUES

64

To conclude this lecture, consider the limiting behavior of the ratios of the record times,

L(n+ 1) Tn = We see that for any x > 1,

n = 1,2,...

L(n)

P{Tn > x} = P{L(n + 1) > xL(n)}

(14.26)

00

_

P{L(n + 1) > xL(n) I L(n) = i}P{L(n) = i} i=n 00

_ E P{L(n + 1) > [xi] I L(n) = i}P{L(n) = i} i=n 00

_ i=n

[i] P{L(n) = i}.

For an integer x, 00

CO

i=n (xi]

P{L(n) = i} _

a

ion xi

00

P{L(n) = i} _

P{L(n) i=n

x

1

x

For an arbitrary x we have 1

< i
x.

1 x L(n) for any x > 1 as n --+ oo. Thus we have proved the following theorem.

THEOREM 14.2. For any x > 1, -+L

lim P{ noo L

()1)

> x} =

-

.

If x > 1 is an integer, then

P{L(n+ 1) L(n)

> x} =

I X

EXERCISE 14.2. Prove the following result by Shorrock (1972b): for a fixed

r = 2,3,... the ratios L(n + k)

'n TA :,n

L(n+k-1)'

k = 1,2,...,r,

are asymptotically independent as n -+ oo. The following exercise is closely related to Exercise 14.2.

15. DISTRIBUTIONS OF RECORD VALUES

65

EXERCISE 14.3. Galambos and Seneta (1975) considered the ratios L(n) and defined integer-valued random variables T(n) by

n=2,3,....

T(n)-1 < L(n(n)1)

2.

Lecture 15. DISTRIBUTIONS OF RECORD VALUES

From record times L(n) we turn now to the record values

n = 1, 2, ... .

X (n) = XL(n) = XL(n),L(n) = ML(n),

It follows from Lemma 13.2 that for any n, m = 1, 2,... the event

=n-1,tm=1} and the random variable Mm are independent. Since

P{Mn < x} = Fn(x), we obtain that (15.1)

P{X(n) < x} = P{ML(n) < x} 00

> P{ML(n) < x I L(n) = m}P{L(n) = m} M=I 00

_ E P{M,n < x I L(n) = m}P{L(n) = m} M=1 00

_ E P{Mm < x}P{L(n) = m} M=1 00

=

r F-(x)P{L(n) = m} = E (FL(n)(x)) = Qn(F(x)), M=1

where

Qn(s) = EsL(n)

is the generating function of the random variable L(n). Now (14.18) implies that (15.2)

(- log(1 - F(x)))k

P{X(n) < x} = 1 - (1 - F(x))

k!

k=O

=J

log(i-F'(x)) vn-I exp(-v) dv

(n - 1)!

,

If

F(x) = 1 - exp(-x),

x > 0,

n=1,2,...

2. IIEC'C)RI) I'INIES AND RECC)R.D \AI.1'1.:S

(i(i

then

(15.3)

P{X (n.) < x}

-J

(n - 1)!

i.e., X (n) has the gamma-distribution with parameter n. An analog of Representation 3.1 for order statistics holds also for record values.

REPRESENTATION 15.1. Let X(1) < X(2) < ... be the record values in a sequence of i.i.d. random variables with continuous d.f. F, and let U(1) < U(2) < ... be the record values related to the uniform distribution on 10, 11. Then for any n = 1, 2, ... the random vector (F(X (1)), ... , F(X(n))) has the same distribution as (U(1), ... , U(n)). The proof of this result is based on the ideas which were used in the proof of Representation 3.1. The basic idea is that the probability integral transformation F(X), which transforms a random variable X with continuous d.f. F into a random variable uniformly distributed on [0, 11, does not affect the ordering of the random variables.

Now we obtain some useful corollaries to Representation 15.1.

COROLLARY 15.2. If F is a continuous d. f. and G is its inverse, then for any n = 1, 2, ... we have (15.4)

(X(1),...,X(n)) `I (G(U(1)),...,G(U(n))).

REMARK 15.3. In contrast to relation (3.2), the equality (15.4) may fail if F

is not continuous. For example, let P{X = 0} = P{X = 1} = 1/2. Then G(s) = inf{x: F(x) > s} = 0 for 0 < s < 1/2 and G(s) = 1 for 1/2 < s < 1. Therefore the vector in the right-hand side of (15.4) has at most two different components, which for n > 2 contradicts the property that X(1) < . . . < X (n). This example relies on the fact that record values may not exist with probability one if the rightmost point of the support of the distribution, x" = sup{x: F(x) < 1}, is an atom, i.e., if P{X = x'} > 0. However, taking a somewhat more complicated distribution than in this example, having, say, a single atom xo such that

P{X<xo}>O, P{X=xo}>0, and P{X>xo}>0, we also see that (15.4) fails.

COROLLARY 15.4. For any n = 1, 2,... the record values X (l) < X(2) < ... and Y(1) < Y(2) < ... corresponding to continuous d.f.'s FI and F2 are related as follows: (15.5)

(X ( 1 ) ,

... , X (n))

d

(H(Y(1)), ... , H(Y(n))),

where H(x) = GI (F2(x)) and GI is the inverse function to FI. Let us mention the following important particular case of (15.5): if X(1) < X(2) < ... are record values related to a continuous d.f. F and Z(1) < Z(2) < ... are exponential record values (related to the standard exponential distribution). then (15.6)

(X(1),...,X(n))

d

(H(Z(1)),...,H(Z(n))).

67

iS. fFS'ITil13l;'I IONS OF RECORD VALI'ES

where H(x) = C(1 - exp(-x)) and C is the inverse function to F. Now we find the conditional distribution

x

cp(x I x1,. .. , x,,) = P(X(n + 1) > x I X(1) = x1, X(2) =

First we show that for any n = 1, 2.... the random variables Yl = XL(n)+1, Y2 = XL(n)+2, ...

are i.i.d. with common d.f. F, and are independent of X (1), X (2), ... , X (n). To this end, observe that for any m the events Cn,m = {L(n) = m}

are determined only by the random variables X1, X2, ... , X,n and do not depend

on Xm+l, Xm+2,... For an arbitrary event B generated by the record values X(1),X(2),...,X(n) and A = {Y1 < xl,...,Yk < xk}, the probability P{AB} can be written as 00

00

P{AB} = E P{ABCn,m} _> P{A I BCn,m}P{BCn,,n} m=0

M=0 00

_ E P{Xm+l <x1,...,Xm+k <xk I BCn.m}P{BCn.m}. M=0

Since the event BC,,,,,, is determined by the random variables X1, X2.... , X,n, which are independent of Xm+l, ... , X,n+k, we see that 00

P{AB} =

, Xm+k < xk}P{BCn,m}

P{X,n+1 < XI, M=0 00

= F(x1) ... F(xk)

P{BCn,m} = F(xl) ... F(xk)P{B}, m=0

which proves the above assertion. Next, the random variable X(n + 1) can be written as XL(n)+T(x(n)) or Yr(X(n)), where -r(u) denotes the index of the first random variable in the sequence Y1, Y2, ... exceeding u. Taking into account the above assertion we obtain that co(xI

x I X(1)

= =

=xn}

x I X(1) = xl,..., X(n) = xn} x} = P{Yr(x..) > x}.

Finding the distribution of the first random variable in the sequence Y1, Y2.... exceeding xn we obtain

Ax I xl,...,xn) = P{Yr(x.,) > x}

=P{Yl > x}+P{Yl <xn,Y2 > x}+... +P{Yl <xn,...,Yk_1 <xn,Yk>XI + _ (1 - F(x)) +F(xn)(1 - F(x)) +... +Fk-1(xn)(1 - F(x)) +..

_

1 - F(x) 1 -

F(x)

,

x>xn.

2. RECORD TIMES AND RECORD VALUES

68

Therefore (15.7)

P{X(n + 1) > x I X (1) = xl, X (2) = X2, ... , X (n) = xn}

_ 1 - F(x) 1-F(xn)'

x>xn,

and the conditional density function fn+1(x I x1, ... , xn) of X (n + 1) given that X(1) = x1, X(2) = x2i ... , X (n) = xn has the form (15.8)

n),

F()

fn+1(x I x1,...,xn) = 1

x > xn,

provided the underlying distribution has density f. It follows from (15.7) that X(1), X(2).... form a Markov chain, and (15.8) implies that the joint density function fn(x1, ... , xn) of the record values X(1), X(2), ..., X (n) is (15.9)

fn(xl,.,xn) = fn(xn Ix1,...,xn-1)fn-1(x1,...,xn-1) f(xl)f(x2)...f(xn)

(1 - F(x1)) (1 - F(x2)) ... (1 - F(xn-1)) = R(x1)R(x2) ... R(xn) (1 - F(xn)) for x1 < x2
n}

PROOF. The relation (16.1) follows from the equalities

P177 1} = P{X1 = n} + P{X1 < n, X2 = n} + P{X1 < n, X2 < n, X3 = n} + .. . = P{X = n} (1 + P{X < n} + P2{X < n} + ... )

P{X=n}

P{X=n}

1 - P{X < n}

P{X > n}

Since the indicators ?70, 77, take only two values, for the proof of their independence

it suffices to show that for any r = 2, 3, ... and for all 0 < a(1) < a(2) < ...

= P{ria(1) = 1,17x(2)

(16.2)

= 1, ... 77.(r) = 1}

H

k=1

P{X -> a(k)j'

Let M(r - 1) denote the time when the record value equal to a(r - 1) occurs. Then (16.3)

P{%(1) = 1,rla(2) = 1,...,7 (r) = 1} = E P{7la(1) = 1 , 77«(2) = 1 ,--- ,7 7 . ( r ) = 1, M(r - 1) = m} m 00

= E E P{ 7r7.(1) = 1, 77a(2) = 1, ... , 77a(r-1) = 1, M(r - 1) = in, in s=1

Xm+1 < a(r - 1), ... , Xm+s-1 < a(r - 1), Xm+s = a(r) } 00

_ E j:P{17a(1) = 1777a(2) = 1, ... ,77 (r_1) = 1, M(r - 1) =m) m s=1

x P{X,,,+1 < a(r - 1)} ... P{X,,,,+s-1 < a(r - 1)}P{X,,,+s = a(r)} _ E P{77x(1) = 1, rla(2) = 1, ... 7 77a(r-1) = 1, M(r - 1) = m} in cc

x E P{X,+1 < a(r - 1)} ... P{Xm+s-1 < a(r - 1)}P{Xm+s = a(r)} s=1

_

1,77x(2) = 1,...,17«(r-1) = 1, M(r -- 1) = m} in

=

P{ 71x(1)

P{X = a(r)} > a(r- 1)}

= 1,77a(2) = 1,...,r7a(r-t) = 1}P{X

P{X = a(r)} P{X > a(r - 1)}

16. SHORR.OCK'S REPRESENTATION

71

Using (16.3) successively we obtain (16.2). Theorem 16.1 implies the following useful result. REPRESENTATION 16.2. Let X1, X2,... be i.i.d. random variables taking non-

negative integer values, and let X(1) < X(2) < ... be the corresponding record values. Then for any m = 0,1, ... and n = 1, 2, .. . (16.4)

P{X(n) > m} = P{rro +'r71 + ...

+77m
m} The relations (16.4), (16.5) can be used for obtaining exact and limiting distributions of suitably centered and normalized record values X(n). We will not formulate general theorems and restrict ourselves to the following example.

EXAMPLE 16.3. Consider the geometric distribution supported on the set 11,2.... }. Thus, let

P{X = n} _ (1 - p)pn-1,

n = 1, 2, .. .

Then

P{X>n}=pn-1 and P{77,,=1}=1-p,

n=1,2,...,

i.e., the random indicators 771,772.... are not only independent, but also identically distributed. In this case the sum

Sm=771+ +llm has the binomial distribution with parameters m and (1 - p), and min(m,n-1)

P{X(n) > m} _ >

(rn)(1

- p)rpm-r.

r=o Since

ETlr = 1 -p and Var?7r = p(1 - p),

r = 1, 2, ... ,

we see that the random variable

Sm,-m(1-p) (mp(1 -p))"2 has asymptotically the standard normal distribution. Then the random variable

(1 - p)X(n) - n (np) 1/2

will be also asymptotically normal. Indeed, using (16.4) and denoting x(np)1/2 m(x)= n + 1 -p

72

2

C)RI) C!\IE5 AND RECORD VALI ES

(assume for simplicity that m(:r,) is an integer) we obtain that for any fixed x

P{

(1-p)X(n)-n >x }=P( X(n)> n+x(np)1/2 _ 1-p (np)1/l m(x)(1 - p) n - m(x)(1 - p) f P (m(x)p(1 - p))

1 /2

PJS,n(x)-m(x)(1-p) I

< (m(x)p(1 - p))
n - 1

(16.6) P{X(n+ 1) = j I X(n) = k} = P{?7k+I = 0,'qk+2 = 0, ... , 77j_1 = 0, rlj = 1}

_P{X>k+1}...P{X>j-1}P{X=j} _ P{X=j} P{X > k + 1} ... P{X > j - 1}P{X > j}

P{X > k + 1}

and

(16.7)

P{X(n+ 1) > j I X(n) = k} =

P{X > j}

P{X>k+1}'

EXERCISE 16.1. Show that for i.i.d. random variables X1i X2, ... taking nonnegative integer values the joint distribution of the record values X(1), X(2).... is given by the equalities (16.8)

P{X(1) = i1iX(2) = i2,...,X(n) = in I P{X = ir} = P{X = in} n_ 11 P{X > jr}' r=1

0 < i1 < i2 < ... < in.

EXERCISE 16.2. Show that if X1, X2r ... are random variables as in Exercise 16.1, then the record values X(1) < X(2) < ... form a Markov chain with transition probabilities given by (16.6).

Now we state one more interesting relation for discrete records. Let 00

pk=P{X=k}, qk=P{X>k}=Ep,, j=k

k=0,1,...,

Iti. SIIORIIO('K'S REPRESENTATION

-:1

and Ti

n=0,1,....

A(n) =1:Lr, r=0 9r

Consider the conditional expectations

E (A(X(n + 1)) 1 X(1),...,X(n - 1), X(n)). Taking into account Corollary 16.4 and Exercise 16.1 we obtain (16.9)

E (A(X(n + 1)) 1 X(1),. .. , X (n - 1), X(n) = m) = E (A(X(n + 1)) 1 X(n) = m) 00

A(k)P{X(n+ 1) = k I X(n) = m} k=m+1

P{X = k} _ E0"A(k)P{X > m} k=m+1

E

k

00

1: Pk

k=m+1

m _r--±

00

Pk Pr r=0 4r k=max(m+l,r) qm+1

r=0 9'R'+1 00

00

Pk

+-0 Qr k=m+1 4m+1

m

+

00

Pk

r=m+1 4r k=r 9m+1

m

00

rr=O Qr

Prr

Pr r=m+1 9m+1

Er+1=A(m)+1.

r=0 4r

Putting

T(n) = A(X(n)) - n, we can rewrite (16.9) as

E(T(n+1) 1 X(1),X(2),...,X(n)) =A(X(n)) - n = T(n) or (16.10)

E (T(n + 1) 1 T(1),T(2),...,T(n)) = T(n),

n = 1,2....

The reader familiar with the theory of martingales, say, on the level of Shiryaev (1980), can observe from (16.10) that the random variables

T(n) = A(X (n)) - n,

n = 1, 2, ... ,

form a martingale with respect to the sequence of Q-algebras

.In = a(X(1),X(2),...,X(n)).

2. RECORD TIMES AND RECORD VALUES

7-1

As a consequence, we obtain that

ET(n) = ET(n - 1) _ = ET(1) = EA(X(1)) - 1 = EA(X1) - 1 00

k

00

= EA(k)P{X

=k}-1=Epk>Pr

k=0

k=0

00

00 =E-r

-1

r=0 qr

00

1: pk-1=1: pr-1=0

r=0 qr k=r

r=0

and

EA(X(n)) = n,

(16.11)

n = 1,2,... .

Turning again to Example 16.3 we see that A(n) = n(1 -p) and hence (16.11) for the geometric distribution has the form

(1-p)EX(n)=n and

E X(n) =

(16.12)

n p,

n = 1, 2,... .

1

Now we state one more important result for the geometric distribution as in Example 16.3. For this distribution (16.8) becomes

P{X(1)=itiX(2)=i2,...,X(n)=tin}

(16.13)

=(1-p)np`n-n

and

(16.14) P{X (1) _ il, X (2) - X (1) = i2, ... , X (n) - X (n - 1) = in} p)npil+...+in-n = = (1 P{X = i1}P{X = i2} .. -P{X = in}, which implies the following theorem. THEOREM 16.5. Let X1, X2, 1, 2, ... with probabilities

pk = P {X j = k} = (1 -

... be i.i.d. random variables assuming values p)pk-1

j=1,2_.,

k = 1, 2, ... .

Then the random variables X(1), X(2) - X(1), X(3) - X(2).... are mutually independent and have the same geometric distribution as the original random variables X1, X2, ... . Let us point out that (16.13) can be easily derived without recourse to formula (16.8). Indeed,

P{X(1) = i1i X(2) = i2, ... , X (n) = in} 00

00

1: ... = P{Xl =i1,X25it,...,Xk2 7n . Obviously. Representation 16.2 fails if 77n are replaced by 77n and X(n) by weak record values.

In order to express the distributions of the X,(n) in terms of distributions of sums of independent random variables we will need other random variables than indicators 771 n.

2. REC'ORU I'IAIF.S AND RECORD VALUES

76

Let µ,,, n = 0, 1,. .., be the number of those weak records in the sequence X1, X2, ... that are equal to n. The proof of the following result can be found in Stepanov (1992).

THEOREM 16.8. The random variables µo, Al, A2.... are mutually independent and

P{µn=m}=(1-rn)r;l,

(16.17) where

rn

n=0,1,..., m=0,1,...,

_P{X=n} P{X > n}

Note that

P{Tjn = 1} = P{µn > 0}. The independent random variables 1U0,µ1,µ2, ... will be used in the following representation. REPRESENTATION 16.9. For any n = 1, 2.... and m = 0, 1, .. .

P{X,,,(n) > m} = P{µo + µl +

+ µn < n}.

EXERCISE 16.3. Find the distribution of the weak record value X.(n) for the geometric distribution as in Example 16.3.

Lecture 17. JOINT DISTRIBUTIONS OF RECORD TIMES AND RECORD VALUES

There are a number of useful relations between the distributions of the record

times L(n), record values X(n), and inter-record times 0(1) = L(1) = 1, 0(n) _ L(n) - L(n - 1), n = 2, 3, .. . Consider the joint distributions of the random variables L(n) and X(n). It is easily seen that for any xi,...,xn and 1 = k(1) < k(2) < --- < k(n)

(17.1) P{X(1) <x1,X(2) <x2i...,X(n) <xn, L(1) = 1,L(2) = k(2),...,L(n) = k(n)}

= P{X1 < xl, max(X2.... , Xk(2)-1) X1 < Xk(2) < X2, - . maX(Xk(n-1)+l.... , Xk(n)-1) Xk(n-1) < Xk(n) < xn} -

.c

...

e

x

h(ul,... , un) dF(ul)... dF(un),

00

where

n-1

h(ul,. . ., un) =

F k(r+l)-k(r)-1(ur)

r=1

if -oo < ul
r I L(n, k) = m} = P{L(n + 1, k) > r L(n, k) = m, L(n - 1, k) = L(2, k) = m2, L(1, k) = k} 0,...,£r(k) = 0} =

- (m+l-k)...(r-k-1)(r-k), r>m>n+k-1, (m+1)...(r-1)r

and

(19.8)

P{L(n + 1, k) = r I L(n, k) = m} 0,...,er-1(k) =

= 0,er(k) = 1}

- (m+1-k)...(r-k-1)k r>m>n+k-1, (m + 1)...(r - 1)r

which corresponds to formulas (14.3) and (14.4). With the record times L(n, k) one can naturally associate the random variables N(n, k) defined as the number of the kth records in the sequence X1, X2, ... , Xn. We have for any k, n, m = 1, 2, .. . (19.9)

P{L(n, k) > m} = P{N(m, k) < n}

and

(19.10)

P{L(n, k) = m} = P{N(m - 1, k) = n - 1, N(m, k) = n}.

The equalities (19.9) and (19.10), like (14.9) and (14.10) related to k = 1, enable us to express the distributions of the kth record times in terms of the distributions of the sums (19.11)

of independent indicators and to use the theory of summation of independent random variables. The relations (19.9) and (19.10) can be rewritten as (19.12)

P{L(n, k) > m} = P{G(k) +

+ ,n(k) < n}

11). THE, kT11 ItEC OHI)S

S. I

and

(19.13)

P{L(n, k) = m} = P{N(m - 1, k) =,n - 1, ,n(k) = 1} =

P{G(k)+

Now we state some asymptotic relations for N(n, k) and L(n. k). To this end remember that the record indicators G (k), G+1(k), ... are independent and observe

that k

and

k k2 Varfn(k)=--z,

n=k,k+1,....

Hence

=k

(19.14)

m=k,k+1,....

m, m=k

and

EN(n,k)=k(logn-

(19.15)

k-i

E

1

-m

n1-),

+y)+O(\

n - oo,

m-1

where -y = 0.5772... is the Euler constant. Accordingly, (19.16)

Var N(n, k) = E N(n, k) - kz

E zl m=k

k-1 1

k6 2

M=1 1

+ D(-), D, since

00

lz 6z = m 6 m=1

m

mz

m=1

n --+ oo,

00

and

k-1

Tr2

0<E

lmz
1 and B (a, b) = r(a)r(b) = fo 1 x°-1(1- x)6-1 dx. r(a+b)

Moreover, let Fnk) denote the v-algebra generated by the random variables L(1, k), .... L(n, k), or equivalently, by the random variables U(1, k), ... , U(n, k). Note

that .F(k) C

.k) C .. .

THEOREM 20.2. For any fixed k = 1, 2.... and -y > 1 the sequence of random variables U(n, k), n > 1, is a martingale with respect to the sequence of o-algebras F(k) Furthermore, (20.2)

E (U(n, k) I U(m, k)) = U(m, k),

n > m > 1,

and (20.3)

E U(n, k) = E U(1, k) = yB (k + 1, k(y - 1)),

n > 1.

20. GENERATING FUNCTION OF THE kTH RECORD TIMES

87

PROOF. Let b = k-y - k. Under the conditions of the theorem, 6 > 0. For the proof it suffices to show that (20.4)

E (B (L(n, k), 6) L(n - 1, k) = m, L(n - 2,k),. .. , L(1, k))

- B(m+1,6), y

m> k,

because (20.4) implies that

E (U(n, k) I '-(k!1) = E (U(n, k) I U(1, k), ... , U(n - 1, k)) = U(n - 1, k),

so that the sequence U(n, k), n > 1, is a martingale. Denote Mk(m,) =

r(m + 1)

r(m-k+1)'

It follows from (19.8) that

P{L(n, k) = r I L(n - 1, k) = m, L(n - 2, k), ... , L(1, k) } = P{L(n, k) = r I L(n - 1, k) = m}

-k(r-k-1)!m! _ r!(m - k)!

kMk(m) Mk+1(r)'

n+k-1<m 1,

-

2. RECORD TIMES AND RECORD VALUES

88

is a martingale for any fixed k = 1, 2.... and arbitrary y > 1. The equality

n > m > 1,

E (U(n, k) I U(m, k)) = U(m, k),

holds because L(n, k), n = 1, 2, ... , is a Markov chain, hence the sequence U(n, 1), U(n, 2),... is a Markov chain as well. REMARK 20.3. The restriction -y > 1 in Theorem 20.2 is only due to the factor

r(ky - k) in the definition of U(n, k). This factor does not depend on n and one can easily check that the sequence n r(L(n, k) + 1) n> y r(L(n, k) + ky - k + 1)' is a martingale for any positive y. In particular, taking

k-/3

)3 m > 1,

E (TT()3) I Tm(/3)) = Tm(f),

and

(k - /3)r(k + 1)

ETn(/3) = ET1(Q) _

(20.9)

kr(k-/3+1)

Now we proceed to the proof of the main result of this lecture. PROOF OF THEOREM 20.1. We obtain from (20.3) that

EUn=ynEB(L(n,k)+1,ky-k)=yB(k+1,k-y-k)=B (k,ky-k), or, putting a = ky - k, (

(20.10)

E B (L(n, k) + 1, a) _ k + 1) \

n

f xa-1(1 1

Relation (20.10) implies the equality (20.11)

J

x)k-1 dx,

1 xQ-1E ((1 - x)L(n'k)) dx = ( k

+

1) -n

f xa-1(1 - x)k-1 dx,

which can be rewritten as e-avE ((1

(20.12)

- e-v)L(n,k)) dv

0

0o

a-«v

knvn-le-kv

o0

dv

(n - 1)! Jo By properties of the Laplace transform, the function o

r(v) = E ((1 -

a > 0.

0

e-)L(n,k))

e-v)k-1 dv.

89

21. NMONIENT CHARACTERISTICS

is the convolution of the functions

ri(v) = kn(n -ll)!kv

r2(V) = (1 - e-' )k-1,

and

v > 0.

Hence we finally obtain that

Qn.k(3) = EsL(n'k) = r(- log(1 - 3)), where

r(v) =

kn

v

(n-1)!Jo xn-1e-kx(1 -

e-(V-s))k-I dx.

0

Thus the proof of Theorem 20.1 is completed.

Lecture 21. MOMENT CHARACTERISTICS OF THE kTH RECORD TIMES

We have shown in Lecture 13 that E L(2) = oo. Therefore, EL(n) = oo for n > 2. This property of the classical record times substantially restricts their use in various statistical procedures. It turns out that E L(n, k) < oo for any n = 2, 3, ... already for k > 1, while for k > 2 all variances Var L(n, k) are finite. In this lecture we will derive a number of formulas for moments of the random variables L(n, k). Let us turn to relation (20.9), which holds for p < k:

ETn()3)=ET1()3)= (k (k k

r(L(n,k)-0+1)

Recall that the gamma-function r(s) satisfies the relation

r(3 + 1) = sr(s).

Therefore, if we assume that Q in (20.9) is an integer r, r = k - 1, k - 2, ... ,1, w get the relation (21.1)

(k - r)n-lE (L(n, k)(L(n, k) - 1) ... (L(n, k) - r + 1)) = k

r(k + 1)

r(k-r+1)'

Using the notation

Mk(m) -

r(m+1) r(m - k + 1)

=m(m- 1)...(m-k+1),

we can rewrite (21.1) as (21.2)

m(r, k, n) = E M,. (L(n, k))

k

_ k - r)

n

1

r(k+1) r(k - r + 1)

k- (k - 1)! (k - r)n(k - r - 1)!'

This equality provides an expression for factorial moments m(r, k, n) of order r = k - 1, k - 2, ... ,1 of the random variables L(n, k) fork > 1 and any n = 1, 2, ... . Let us state some particular relations which follow from (21.2).

2. RECORD TIMES AND RECORD VALUES

90

COROLLARY 21.1. The following equalities hold

m(1, k, n) = E L(n, k) =

(21.3)

kn

(k -

k >- 2,

n >- 1,

1)n-1I

m(2, k, n) = E L(n, k) (L(n, k) - 1) _ (kn (k 2)nl i ,

EL 2 (n, k) = m(2, k, n) + m(1, k, n) = (k (2)nl)) + (k k1)n-1

'

(21.4)

Var L(n, k) = m(2, k, n) + m(1, k, n) - m2(1, k, n)

_

kn

2n

n

n > 1, k > 3.

kl)n-1 - (k = (k (2)nl 1 + (k As is well known, the factorial moment of order r = 1, 2.... of a random k1)2n-2'

variable l; is defined to be

r(c-r+l)/ For a nonnegative random variable l; we will define factorial moments of negative order by means of the same relation:

ur(n) = E

r

E ((£ + 1) ... ( - r)

1

r +)1)

Now we can consider the factorial moments

m(r, k, n) = E Mr.(L(n, k)) = E ( E

r(L(n, k) + 1) l r(L(n, k) - r + 1)

1 1 ((L(n,k)+1)...(L(n,k)-r)l

of negative order r for the kth record times L(n, k). Since (20.9) continues to hold for negative 0, we can employ (21.1) for r = -1,-2.... to derive for these r the following formula for the moments m(r, k, n): k! k In-1 k )n-1 r(k + 1) _ m(r, k, n) _ (k --r)! r(k-r+1) =

- (k-r/

(k-r

In particular, for k > 1 and n > 1 we obtain (21.5)

(k + 1)n'

_

1

(21.6)

kn-1

1

E (L(n, k) + 1 / E ((L(n,k) + 1)(L(n, k) + 2)I

kn-' (k + 2)n (k + 1)'

Using the property (20.8) of the random variables Tn()3) _

(kn 8)nr(L(n, k) + 1) k r(L(n, k) -,3 + 1) '

we can find the product moments of the random variables L(n, k). For example, if n > m and k > 1 we can argue as follows.

21. MOMENT CHARACTERISTICS

91

Obviously, the kth record times are related to Tn(;3) by the equalities L(n, k) =

k" 1)nT,l(1)

(k -

km

and

L(m, k) _ (k - 1)-

Using the martingale property (20.8) we obtain that (21.7)

kn+m

E L(n, k)L(m, k) = (k-1)n+mETn(1)Tm(1) kn+m

(k - 1)n+mETm(1)E (Tn(1) I Tm(1)) kn+m

kn+m

(1)

(k - 1)2m

"_

(k-1)n+mETm= (k-1)n+m k2EL (m'k) kn-m

(k - 1)n-m

km km(k - 1) (k - 2)m-1 + (k - 1)m-1

Now (21.3) and (21.7) imply the following expression for the covariance between the kth record times: kn m km (k _ (21.8) cov (L(n, k), L(m, k)) _ (k- 1)n-m (k (2) m-1 + (k k1)m-1) kn+m

n>m>1.

(k-1)n+m-2'

EXERCISE 21.1. Using the martingale property (20.8) as it was done in the proof of (21.8), show that for k > 1 and any n1 > n2 > n3 > n4 the following equality holds for the covariances between ratios of the kth record times: L(nlik) L(n3,k)

cov

( L(n2i k)' L(n4, k)) =

0.

EXERCISE 21.2. Let the sequence f (n), n = 0,1, ... , be defined as

f(0)=0 and f (21.9)

n = 1, 2, ... and k = 1, 2.... the following equality holds: nk 1 E f (L(n, k)) = f (k) +

REMARK 21.2. Since

logn=f(n)-7-2n+O\n 1

= f(n) - 7 - 2(n+ 1) +O((n+ 1)(n+2)),

n

---+ 00,

where 7 is the Euler's constant, formulas (21.4), (21.5), and (21.8) can be used for finding logarithmic moments E log L(n, k). For example, we have (21.10)

E log L(n, k) = f (k) + n

1

-7

kn-1

2(k+ 1)n

+O((k+2/ )'

n

oo.

2. RECORD TIMES AND RECORD VALUES

92

Relations of the form (21.10) are important because it is the logarithm of L(n, k), rather than L(n, k) itself, which is asymptotically normal.

Lecture 22. TATA'S REPRESENTATION AND ITS GENERALIZATIONS

Now we pass from the kth record times to the kth record values X(n, k) related to a sequence of i.i.d. random variables X1, X2, ... with continuous d.f. F. Denote by Z(n, k) the kth exponential record values related to a sequence of i.i.d. random variables Z1, Z2, ... with standard exponential distribution. One can easily check that using the probability integral transformation (as it was done for ordinary record values, see Corollary 15.4) we obtain the following important assertion, which will allow us to restrict ourselves to the exponential distribution when dealing with the kth record values.

REPRESENTATION 22.1. For any n = 1, 2,... and k = 1, 2,... we have (22.1)

(X(1,k),...,X(n,k))

d

(H(Z(l,k)),...,H(Z(n,k))),

where H(x) = Q(1 - exp(-x)) and Q is the inverse function to F. In Lecture 15 we stated Tata's representation (see Tata (1969)) for exponential record values Z(n) (see Representation 15.6) which implies that the random variables Z(1), Z(2) - Z(1), Z(3) - Z(2) are independent and have the standard exponential distribution. As an important consequence of probability integral transformation and Tata's representation we get a possibility to express the distributions of arbitrary record values in terms of distributions of sums of i.i.d. exponential random variables. This is given by the following theorem. THEOREM 22.2. We have (22.2)

{X(n)J°°

1

=

where H(x) = Q(1 - exp(-x)), Q(x) is the inverse function to F(x), and W1, W2.... is a sequence of i.i.d. random variables with standard exponential distribution. Dziubdziela and Kopocinsky (1976) obtained the following generalization of Tata's result. THEOREM 22.3. Let Z1, Z2, ... be i.i.d. random variables with exponential d.f.

G(x) = 1 - exp(-x),

x > 0,

and let Z(n, k), n = 1, 2, ... , be the corresponding kth record values. Then for any

k=1,2,... (22.3)

{Z(n,k)}0o

n=1

d { w1 + ...+wn}°o k

n=1'

where W1iW2,... are i.i.d. random variables with standard exponential distribution. Relation (22.3) implies the following generalization of Theorem 22.2.

22. TNI'A'S iEPRESENTATION

THEOREM 22.4. For any k = 1, 2,... {H(w1+...+wn/Jx

{X(n)j0'n=

(22.4)

ll 11

k

n=1

We can derive from (22.4) a number of useful relations.

THEOREM 22.5. The sequence X (1, k), X(2, k),... is a Markov chain, and for

anyk>1, n> 1, andx>u (22.5)

P{X (n + 1, k) > x I X (n, k) = u} =

(1 - F(x) k 1 - F(u))/

PROOF. Theorem 22.4 implies that (22.6)

P{X(n+1,k)>xI X(n,k)=u,X(n-1,k),...,X(l,k)} _us

x I H(w1+

H(w1+

k+wn_1l/)...,H(k

-klog(1-F(x)) -k log(1 - F(u)), w1 + - + wn_ 1, ... , w1 }. The sequence w1, w1 + w2, ... is a Markov chain and

+wn=v}=P{wn+1>u-VI = exp{-(u - v)},

u > v.

Hence the right-hand side of (22.6) equals the conditional probability P{w1 +

+wn+1 > -klog(1 - F(x)) I w1 +

+wn = -klog(1 - F(u))}

and has the form

exp {k(log(1 - F(x)) - log(1 - F(u)))

1-F(x)lk

(1 - F(u)

This proves the theorem.

Consider now two sequences of i.i.d. random variables: X1, X2i ... with continuous d.f. F and

Y1 = min{X1i... , Xk}, Y2 = min{Xk+1,... , X}, .. .

with d.f. G(x) = 1 - (1 - F(x))k. Let X(n,k) denote, as before, the kth record values related to the sequence X1i X2,..., and let Y(n, 1) denote the ordinary record values (k = 1) in the sequence Y1, Y2, .... The following theorem relates the distributions of X (n, k) and Y(n, 1). THEOREM 22.6. For any k = 1, 2, ... , 0o

3{X(n,k)}1 n= = {Y(n,1)}n=1.

2. RECORD TIMES AND RECORD VALVES

94

PROOF. It suffices to observe that the function G- (x), inverse to G, has the form G`(x)=Q(1-(1-x)Ilk),

where Q(x) is the inverse function to F. Then the application of Theorem 22.4 with k = 1 and

H(x) =G'-(1-exp(-x)) =Q(1-exp(- -k yields the expression Q(1 -exp (

- Wl + . k . +Wn ))

in the right-hand side of (22.4), which means that Y(n,1) and X(n, k) have the same distribution.

REMARK 22.7. The relationship between the distributions of the kth and ordinary record values stated in Theorem 22.6 enables us to write down immediately the d.f. of X(n,k). Since we know that

- log(1-F(z)) P{X(n,1) < x} = (n

1

1)I fo

un-1e-,. du,

we have to substitute G(x) = 1 - (1 - F(x))k for F(x) in the right-hand side of the equality. Hence we obtain that for any k > 1 and n > 1 k log(1-F(z))

(22.7)

P{X (n, k) < x} =

un-1e-° du.

1

(n - 1)!

o

Consider now the sequences of random vectors X (n)

= (XL(n,k)-k+1,L(n,k),

,

XL(n,k).L(n,k))

constructed for the sequence XI, X2,..., and Z(n) = (ZL(n k)-k+1,L(n,k),

...

TL(n,k),L(n,k)),

n

related to i.i.d. standard exponential random variables ZI, Z2, .... Using the probability integral transformation one can easily check that the vectors X(n) and (H(ZL(n,k)-k+1,L(n,k)),

, H(ZL(n,k).L(n,k)))

have the same distribution. Note that Dziubdziela and Kopocinsky proved that X (1), X (2).... form a Markov chain. Now we will formulate without proof results by Ahsanullah and Nevzorov (1996) describing the structure of order statistics generated by the kth records. THEOREM 22.8. Let

WI,k-I :5 ... < Wk-,,k-1

and WI k < ... < Wk.k

be the order statistics related respectively to the sets

W1,W2,...,wk_1

and W1,W2,...,Wk

23, CORRELATION COEFFICIENTS FOR RECORDS

95

of i.i.d. random variables having the standard exponential distribution. Then the (2k - 1)-dimensional vector (ZL(n,k)-k+2.L(n,k) -ZL(n,k)-k+1,L(n,k), .. , ZL(n.k).L(n,k) -ZL(n,k)-k+1.L(n.k), ZL(n+1,k) -k+1,L(n+1,k) - ZL(n,k)-k+l,L(n,k),

,

ZL(n+l,k),L(n+l.k) -ZL(n,k)-k+1,L(n.k))

is independent of ZL(n,k)-k+1,L(n,k) and has the same distribution as (w1 ,k-1,

,

wk-1,k-1, w1,k, ... , wk,k).

One can deduce from Theorems 22.3 and 22.8 the following assertions for exponential order statistics: ZL(n,k)-k+1,L(n,k),

,

ZL(n.k),L(n,k)-

COROLLARY 22.9. For any k > 1 and n > 1

T(n, k) - ZL(n,k)-k+1,L(n,k) +"'+ZL(n,k),L(n,k)

d

wl ++wn+k-1,

where w 1 , . . . , Wn+k_ 1 are i. i. d. with standard exponential distribution, i.e., the sum T (n, k) has the gamma-distribution with parameter n + k - 1.

COROLLARY 22.10. Let

S(n, k) = T (n, k) - kZL(n,k)-k+1,L(n,k),

where T(n, k) are defined in Corollary 22.9, and V(n) = S(n + 1,k) - S(n,k).

Then for any k = 1, 2.... the random variables V (n), n = 1,2,..., are i.i.d. with standard exponential distribution, while the random variable S(1, k) is independent of V(1), V(2),... and for k > 1 has the gamma distribution with parameter k - 1.

Lecture 23. CORRELATION COEFFICIENTS FOR RECORDS

We know from (21.7) and (21.4) that for k > 3, cov (L(n, k), L(m, k)) =

k'

(k - 1)n-m

km km(k - 1) (k - 2)m-1 + (k - 1)m-1

kn+m

(k -

1)n+m-2'

+

>m>

and

kn(k- 1) + kn - k2n , n (k - 2)n-1 (k - 1)n-1 (k - 1)2n-2' These equalities enable us to evaluate the correlation coefficients between the kth VarL(n,k) =

record times.

Now we discuss how to obtain moments of the kth record values X(n, k) and, in particular, of the classical record values X(n). The simplest calculations are needed in the case of the exponential distribution, where

F(x) = 1 - exp(-x),

x > 0.

2. RECORD TIMES AND RECORD VALUES

96

We can use the representation (22.3): oc

{Z(n,k)}n=1

d Iwl l

k

J)n=1'

where w1,w2i ... are i.i.d. random variables with standard exponential distribution. Since

E wn = 1

and

Var wn = 1,

we obtain that

E Z(n, k) = k ,

n = 1, 2, ... ,

Var Z(n, k) = T2 ,

m < n.

cov (Z(m, k), Z(n, k)) = Var Z(m, k) = k2

Hence the correlation coefficient p between Z(m, k) and Z(n, k) does not depend on k and is given by

p = p(Z(m, k), Z(n, k)) =(ml (n)

(23.1)

1/2

m < n.

For other d.f.'s we can use representation (22.4):

{X(n,k)}°° n=1

11

n=1

k

where H(x) = Q(1 - exp(-x)) and Q(x) is the inverse function to F(x). Then

EX(n,k) = EH(Z(n,k)) = EH(k ) and

VarX(n,k) = Var H(Z(n, k)) =VarH(k ), where 77, = w1 + (23.2)

E X (n, k) =

+wn has the gamma distribution with parameter n. Therefore

I'

f'H (x)

exp(-x)xn-1 dx

(n-1)!

k

00

kn

(n - 1)! J0

H(v)

exp(-kv)vn-1

dv

kn

(n - 1)!

Jo

- (nkn- 1)! (nkn - 1)!f

Q(1 - exp(-v)) exp(-kv)vn-1 dv

Q( z )( 1-z )k-1 (- l0g( 1-z))n-I dz

u(1 - F(u))k-I ( - log(1 - F(u)))n- dF(u),

in particular, (23.3)

EX(n) = f

00 u( - log(1 - F(u)))n-1 dF(u) (n - 1)! 00

23. COI?.RELATION COEFFICIENTS FOR RECORDS

07

and similarly, (23.4)

H2 (x) exp(-x)xr-1 dx

E X2 (n, k)

(n-1)!

k

fo

00 F(u)))n-1

u2(1 - F(u))k-1( - log(1 -

(nkn1)I

dF(u)

and

u2(- log(1 - F(u)))i-1 dF(u)

EX2(n) _

(n - 1)!

Note also that for m < n E X (m, k)X (n, k) = E H (Z(m, k)) H(Z(n, k)).

Since kZ(m, k) and k(Z(n, k) - Z(m, k)) are independent and have gamma distributions with parameters m and n - m respectively, we obtain (23.5)

f

EX(m,k)X(n,k)

-

H(U)H

Jo

(+V exp(-(u +

kH\

k

/

V))um-lvn-m-1 dudv

(m - 1)!(n - m - 1)!

Formulas (23.2), (23.4), and (23.5) enable us to evaluate expectations, variances, covariances, and correlation coefficients of the kth record values for specific distributions. Here we will show that the largest possible correlation coefficient p(X (m, k), X(n, k)) between two kth records is attained for the exponential distribution and is equal to (m/n)1/2 if m < n. So, we fix m and n assuming without loss of generality that m < n. We will only consider the d.f.'s F with finite second moments E X2(n, k) and E X2(m, k), i.e., we assume that (23.6)

I(r) =

J

u2(1 - F(u))k-1( - log(1 - F(u)))'-1 dF(u) < cc

for r=mandr=n. EXERCISE 23.1. Show that finiteness of either quantity I (m) or 1(n) does not imply finiteness of the other.

THEOREM 23.1. For any k = 1, 2, ... , m < n, and any continuous d.1 F such that

f0,00 u2(1 - F(u))k-1 ( - log(1 - F(u)))r-1 dF(u) < oo, the following inequality holds: (M) 1/2

(23.7)

p(X (m, k), X (n, k))
0.

This condition means that =bo+bjLi-1(x)-

h1(x) =ao+aiLi -1(x) and h2(x)

Substituting the expressions for the Laguerre polynomials from (23.9) we obtain that (nm) 1/2

Ip(h1(Z(m)), h2(Z(n)))I = if and only if h1(x) = ao + a1(m - x)

and

h2(x) = bo + bl(n - x).

We see that equality in (23.7) holds when

hi(x) = h2(x) = H(x) = Q(1 - exp(-x)), where Q is the inverse function to F. Since

Ip(H(Z(m)), H(Z(n))) I can be equal to (m/n) 1/2 only if H(x) is a linear function, we conclude by solving the equation Q(1 - exp(-x)) = Cl + C2X

that

F(x)=1-exp Thus the proof is completed.

- x- c1 C2

I,

x>cl. 0

21. RECORDS FOR NONSTA'I'IONARY SEQUENCES

101

REMARK 23.2. Note that for the correlation coefficient between order statistics the following inequality holds: (23.23)

/

P(Xi,n, X3,n)

ri(n+1-j)\1/2 \ j(n + 1 - i))

where the equality attains only for the uniform distribution. It may be of interest to note that Szekely and Mori (1985), who proved (23.23), used the Jacobi orthogonal polynomials.

Lecture 24. RECORDS FOR NONSTATIONARY SEQUENCES OF RANDOM VARIABLES

There were a number of attempts to apply the classical scheme to records in

various sports. The most attention in this respect was paid to track events in athletics. In some track and field sports (for example, a mile run) records are available for a period longer than a century and there is a sufficient amount of data for statistical processing. It was shown that in almost all situations the stationary model does not fit the available tables of records. This is by no means surprising

because the methods of training, the sport facilities, the conditions of life and, the competition rules change even within a short period of time. The time of the postman who won the Marathon race in Athens in 1896 is as incomparable with the times shown by the professionals in Atlanta in 1996, each of whom was accompanied by a team of trainers and doctors, as cars at the beginning of the 20th century with contemporary Formula-1 racing cars.

In the literature a number of new record models were proposed to take into account the progress of human abilities. We will describe some of them.

Record models incorporating trend. The simplest (for description) model consists in replacing the sequence of i.i.d. random variables X1, X2.... by Y n = Xn + c(n),

n = 1, 2, ... ,

where c(n) are some constants depending only on n. The most conveniently treated is the case c(n) = cn, where c > 0 when the upper records are considered, and c < 0 for the lower records. This model was thoroughly studied by Ballerini and Resnick (1987). Although the model is comparatively simple, even the proof of asymptotic normality of the number of records N(n) meets certain technical difficulties. The problem is that for c 54 0 the record indicators G = 1{M(n)>M(n-1)},

n = 2, 3, ... ,

are no longer independent, so that the classical limit theorems for independent summands become inapplicable. Furthermore, the probabilities pn = P{G = 1} in this case depend on the underlying d.f. F. Hence one has to restrict oneself to certain classes of d.f.'s F, to show that dependence between t;,, and {,,, decays at a due rate with growing difference n - m, and to apply limit theorems for weakly dependent random variables. It is worth mentioning that while in the classical case E N(n) grows at a logarithmic rate, the best explored cases of the model with trend exhibit a linear growth of the expected number of records.

2. RECORD TIMES AND RECORD VALUES

102

Pfeifer's scheme. The next model of records can also be connected with sports. The German athlete Uwe Hohn on July 20, 1984, was the first to exceed the 100 meter mark in javelin throwing. His javelin landed at 104.8 m, which made this sport dangerous for spectators. Then the design of the javelin was changed by shifting its center of gravity forward, which reduced the length of throws. The records for the new javelin have not yet surpassed Hohn's. But the best throws nowadays come close to 100 m, and undoubtedly the rules will change again as soon as this mark is exceeded. The idea of varying distributions of the random variables in the sequence X1, X2, ... is basic for the scheme by Pfeifer (1982, 1984).

Let {Xnk, n > 1, k > 1} be a double array of independent random variables having in each row Xnl, Xn2i ... a common d.f. Fn. It will be convenient to define first the inter-record times 0(n) = L(n) -L(n-1) rather than the record times L(n). They are defined as

0(1) = 1,

A(n + 1) = min{k: Xn+I,k > Xn.A(n) },

n = 1, 2, ...

.

Then the record times L(n) and the record values X(n) in Pfeifer's scheme are given by

L(n) = 0(1) +

+ A(n)

and

n = 1,2,... . If Fl = F2 = .... then these definitions coincide with the definitions of the classical X (n) = Xn,o(n),

records statistics. Pfeifer obtained a number of results for records in his scheme. In particular, he showed that the vectors (i(n), X (n)), n = 1, 2, ... , form a Markov chain with transition probabilities

P{0(n) = k, X (n) > x 10(n - 1) = m, X (n - 1) = y}

(24.1)

= (1 - Fn(x)) (Fn(y))k-1,

x > y.

It was proved that the sequences (L(n), X (n))

and X (n),

n = 1, 2, ... ,

also form Markov chains, though the random variables L(1), L(2), ... need not possess the Markov property. Note that in Pfeifer's scheme the inter-record times

D(1), ... , 0(n) are conditionally independent given the values of X(1), X(2),..., X (n - 1) and

P{0(1) = 1,i (2) = k(2),..., 0(n) = k(n) I X(1),X(2),...,X(n- 1)} n

_

(1 - Ft(X(i - 1)))(F=(X(i -

k(i) = 1,2,...,

i = 2,...,n.

i=2

EXERCISE 24.1. Using (24.1) and the Markov property of the vectors

(0(n), X (n)),

n = 1, 2, ... ,

show that if

Fn(x)=1-exp{-fin},

x>0,

24. RECORDS FOR NONSTATIONARY SEQUENCES

103

with an > 0, n = 1, 2, ... , in Pfeifer's scheme, then the random variables

T(1) = X(1), T(2) = X(2) - X(1),... are independent and

P{T(n) < x} = Fn(x),

n = 1,2,... .

The scheme of Balabekyan-Nevzorov. One more scheme was proposed by Balabekyan and Nevzorov (1986) and developed by Rannen (1991). Consider again sports competitions, for example, the long jump. Let m athletes of different skill jump in turn, making n attempts each. We can assume that their jump lengths can be described by a sequence of independent random variables

X1,...,Xm,...,Xm(n-1)+1,...,Xmn with d.f.'s

F1, ... , Fm, ... , Fm(n-1)+1, ... , Fmn

such that

k = 1,2) .... n, r = 1, 2, ... , m, Fm(k-1)+r = Fr, i.e., this sequence consists of n repetitions of the group of m d.f.'s Fl,... , Fm. This scheme combines features of a nonstationary model (m different distribution functions) and of the classical records model, since the longest jumps in each attempt Yk = max{Xm(k-1)+l, ..

,

k=1,2,...,n,

Xmk}+

form a sequence of i.i.d. random variables with common d.f. M

G = 11 Fr. r=1

Thus we consider a sequence of independent random variables X1, X2, ... with

continuous d.f.'s Fl, F2.... such that Fm(k-1)+r = Fr,

k = 1, 2, ... .

r = 1 , 2, ... , m,

Let N(nm) denote the number of records in the sequence XI, ... , Xm, ... , Xm(n-1)+1, ... , Xmn, i.e., the number of records counted after n runs each of which consists of m random variables.

THEOREM 24.1. For any n > 1 and m > 1, (24.2)

( m2 l sup IP{N(nm) - logn < x logn} - 4'(x)I : c9 \lognl' X

where c is an absolute constant and

l

! g(x) = max lx1/2, x1/2log r

l / }. 1

2. RECORD TIMES AND RECORD VALUES

104

PROOF. Let Yk = max{Xm(k-1)+1,

,

k = 1,2,... .

Xmk},

The random variables Y1, Y2, ... are independent and have the common d.f. M

G=fFr. r=1

Let 1, e2.... be the record indicators in the sequence Y1, Y2,. .., i.e., {k = 1 if max{Xm(k-1)+1, ... , Xm,k} > max{X1,.

, Xm(k-1) }

Define the variables n = 1,2,...,

NI(n) = b1 + ... + en,

which count the number of "record" runs, so that in each run only one record

achievement is counted. Denote by vk the true number of records in the run Xm(k-1)+1,

, Xmk,

k=1,2,....

Now let 11k = max{0, vk - 1}

and let N2 (n)

=r11+...+77n,

n= 1,2,... .

Then

N(nm) = (6 + ... +

.

) + (771 + ... +'in) = N1(n) + N2(n),

n = 1, 2,... .

Since F1 i . . . , Fm are continuous d.f.'s, G is also continuous. Therefore the indicators

1, £2.... are independent, P{en = 1) = 1/n, and N, (n) - log n (log n) 1/2

has asymptotically the standard normal distribution. Invoking (14.22) we can write (24.3)

sup IP{N1(n) - log n < x log n} -t(x) I
2, n > 1, and r > 1 we have (24.4)

E (771 + ....{- ]n)r < (m -

1)rrr+1

24. RECORDS FOR

SEQUENCES

PROOF. We can write Er = E (771 +

LL

+ 77n )r as

r

Er =

(24.5)

T.

1 k T

T1

k=1

REDk

105

1

... +

F (rla

ii

)

,

where R E Dk means that the summation extends over all vectors

R = (Ti,...,rk) Since O < i, < (m-1),

+ rk = r.

such that Ti > 0,...,rk > O and r1 + i = 1, 2,..., we have

r

(24.6) Er < (m - 1)' k=1 REDk

P{77,, > 0,

TI

r1I

k

77,, > 01-

Let

m=1,2,...,k,

ao=0, /3m=am-am-1-m, G(m1,m2,z) =

fl

G(1,0,x) = 1.

F. (X),

M1 5,95M2

Note that for any 1 < m1 < m2 < n we have 1 - G(m1i m2, x) < 1 - G(X)-

(24.7)

For simplicity of notation, denote v1 = z1 and vl = max{y1-1, z1}. Obviously, (24.8) P{77., > 01 ... , 77ak > 01 m-1 00

00

= f d(Gp1(z1)) E G(1,j1 - 1,v1) 00

j1=1

00

G(1,j2 - 1,v2) f

d(GQ'(z2))

m-

dG(71 + l,m,y1) l 00

00

dFj,(z2) f dG(j2+ 1,m,y2)... I7

v2

12=1

OD

x

Jvt

m-1

x-

100

dFj,(x1)

1

f d(GQ"(zk)) E G(1,jk - 1,Vk) f dFjk(xk) f dG(jk + 1,m,yk) jk=1

xk

vk

The inequalities (24.7) and m-1

m-1

EG(1,j-1,z)(1-Fj(z))=1- fF1(z) < 1-G(z) j=1

!=1

imply that (24.9) m-1

dF,,(xi)

l3,>o

Z'MZOO dG(ji+1,m,yi) r-0 d(GI`(z1))EG(1,j1-1,vi) j,=1 ,

0

M-1

5 E G(1,j, - 1,v1) fmo d( j,=1

1

1--(z,

) f (1 -G(x,))dF,,(x,) ,,

2. RECORD TIMES AND RECORD VALUES

106

( (1 - G(vl))2 \ 1 -G(zi)J 1

00

d( no

(1 - G(z1))2 = 1.

1

1 - G(xi)

Now applying the arguments used for the proof of (24.9), we obtain from (24.6) and (24.8)

E ...E0k>0P{7ja, > 0, ... , 7ja, > 0} < 1,

0,>_0 hence

EE r

Er j} =

n(1)

S(j + 1)

and

P{0(n) > j}

_E E ... 00

00

k1=2 k2=ki+1

00

n(1)n(kl)...n(k,1-1) S(k11) S(kn-1 - 1)S(kn-1 +j)' k_j=k_2+1

j =0,1,...,

n=2,3,....

THEOREM 25.2. For any j = 1, 2, .. . (25.1)

pj = lim P{A(n) = j} = (A - 1)A-l.

Yang used the limiting distributions (25.1) with \ = 1.08 to analyze the frequencies of occurrence of Olympic records between 1900 and 1936. The actual inter-record times turned out to be much shorter than theoretical. Of course, it can be easily explained by different rates of growth of the total world population and the number of athletes, as well as by many other circumstances that enhance the progress in sports. Although Yang's model was not very suitable for the analysis of sport records, it stimulated appearance of new record models. Yang's model originated from a concrete problem related to the sport statistics. This was the reason for taking the d.f.'s F of the underlying random variables X1, X2, ... as Fk = FnM, where n(1), n(2), ... was a specific sequence of integers which formed a geometric progression. It is natural to generalize this scheme by taking arbitrary positive numbers instead of integers n(k). But the main reason for the passage to the F"-scheme is the independence property of record indicators which is characteristic for this scheme.

DEFINITION. A sequence of independent random variables X1iX2,... with d.f.'s F1, F2, ... forms an F"-scheme if F k = F"(k),

k = 1, 2, ... ,

where F is a continuous d.f., and a(1), a(2).... are some positive constants.

25. RECORD TIMES IN THE F°-SCHEME

109

REMARK 25.3. The underlying d.f. F can always be taken to equal Fl. Therefore, without loss of generality, we can assume that a(1) = 1. If all exponents a(n), n = 1, 2, ... , are equal, the F°-scheme reduces to the classical scheme determined by a sequence of i.i.d. random variables.

Consider the record indicators f;1i12..... It turns out that Renyi's Lemma can be carried over with minor modifications to the F°-scheme. Let

n= 1,2,... .

S,6

LEMMA 25.4. In the F°-scheme, the indicators S1, 2, ... are independent and (25.2) P{G = 1} = 1 - PIG = 0}

_

S(n) - S(n - 1)

a(n) a(1) +

n = 1, 2, ... .

S(n)

+ a(n)

PROOF. First we prove the second assertion. Since the d.f. Gn_1 of the maximum M(n - 1) = max(XI, X2, ... , Xn_1) has the form Fs("-1),

Gn-1 = F1F2 ... Fn_1 =

and the random variables Yn_1 and Xn are independent, we obtain that

P{Cn = 1} = P{Xn > max(XI, X2, ... , Xn-1)} =

fx

xLGn_l(x)dFn(x)

'00

_

dF,(n)(x)

= Jo XS(n-1) d(x°(n)) a(n) _ a(n) a(n) + S(n - 1) S(n) Fs(n-1)

1

Now in order to prove independence of the indicators it suffices to show that for any 1 < k(1) < k(2) < . < k(r) we have =

r

II P{ek(m) =1}

P{G(1) = 1, G(2) = 1, ... , k(r) =1}

(25.3)

M=1

_

-

r

a(k(r)) S(a(k(r))) .=1

The probability integral transformation allows us to assume, without loss of generality, that F(x) = x

and

Fn(x) = x°("),

0 < x < 1.

Then (25.4)

l

tt 1,Sk(2) = 1,...,Sk(r) = 1} CC

= P{Xk(I) > )VI(k(1) - 1), Xk(2) > M(k(2) - 1), .... Xk(r) > M(k(r) - 1)}

10U'1

(k(1)-1) d(,ui (k(1)))

l

1

S(k(2)-I)-S(k(1)) d(u

Jug

1

UyS. (k(r)-1)-S(k(r-1)) d(1lr°(k(r)) ).

X

fur-11

l

2))1 1

x ...

2. RECORD TIMES AND RECORD VALUES

110

Successive integration in (25.4) leads to the required expression for the probabilities

P{G(1) = 1,...,G(r) = 1}. The following result is closely related to Lemma 25.4. THEOREM 25.5. Let d.f.'s F1, F2i ... , F, of independent random variables XI, X2i ... , X,, be continuous and

1 < j < n - 1,

0 < Fj (a) < Fj (b) < 1,

(25.5)

for some a and b, oo < a < b < oo. If the vector

and the indicator Sn are independent for any choice of the d.f Fn, then there exist positive

constants a(2),..., a(n - 1) such that

2<jxl we obtain the equality z

(25.7)

Hk+1(x) f Gk-1(y) dFk(y) = c(k)Gk-1(x)Fk(x)Hk+1(x),

where

c(k) = P{£k = 1} = f : Fi(x) ... Fk-1(x) dFk(x). 00

Condition (25.5) implies that

0 a, (25.7) implies

fCk_l(Y)dFk(Y) = c(k)Gk-1(x)Fk(x) = c(k) and (25.8)

f

\

f

LFk (u) dGk-1(u) )

Gk-1(u) dFk(u) + 00

x

Gk-1(u) dFk(u) = 7(k) TOO Fk(u) dGk -I (U),

where 0 < ry(k) < oo,

7(k)

j = 2,3,...,n.

= 1 c(c(k),

We see from (25.8) that

fGk(u)d(logFk(u)) =7(k) f xGk(u)d(logGk-I(u)) J. and

(25.9)

f

x

a

Gk (u) d( log

Fk(u) ) =0 (Gk-1(u))7(k)

x > a.

Since G, (u) > 0 for all u > a, we conclude from (25.9) that Fk(x) = d(k) (Gk-1(x))''(k),

where d(k), k = 2,. .. , n, are some constants. As x tends to infinity we have d(k) = 1, k = 2,3,...,n, (Gk-1(x))1+,(k)

Gk(x) = Gk-1(x)Fk(x) =

=

(G1(W))(1+-y(k))...(1+-y(2))

=

(Fl(x))(1+'y(k))...(1+'y(2))

and (25.10)

Fk(x) = (Gk-1(x))ry(k) =

(FJ(X)),y(k)(I+-e(Ac-I)) --- (1+-t(2)),

k = 2,3,...,n. Thus we have shown that the assertion of the theorem holds for x > a with exponents

c(j) a(j)='Y(j)(1+-t(j-1))...(1+7(2))= (1 - c(2))...(1 - c(j))

j=2,...,n.

By the definition of a there exists k such that Fk(a) = 0. Then (25.10) and continuity of d.f.'s F1,.. . , Fn imply that Fk(a) = 0 for any k, hence (25.10) holds

not only for x > a, but also for x < a. The assertion on independence of the indicators follows by Lemma 25.4.

2. RECORD TIMES AND RECORD VALUES

112

REMARK 25,6, Condition (25.5) means that P{X, < X,} > 0 for any i # j, i, j = 1, ... , n - I. There are simple examples showing that without this condition the theorem may hold not only for the F°-scheme. For example, let Xj have

the uniform U((aj, aj+l]) distribution, j = 1, ... , n - 1, with a1 < a2
n}. Note that for sequences of i.i.d. random variables there was no problem of existence

of record times because S(n) = n in the classical scheme. In general, to deal with records in the F°-scheme we need the restriction on the exponents a(n),

lim S(n) = oo.

(25.12)

n-.oo

As we pointed out, the classical record model can be embedded into the F°scheme. It turns out that the kth record times L(n, k) can also be embedded into the F°-scheme. Indeed, the distributions of these random variables are determined by the sequence of independent indicators .1(k),Wk),... such that 1} = 0

for j = 1,...,k- 1, P{ek(k)=1}=1, and

j>k.

25. RECORD TIMES IN THE F'°-SCHEME

113

Now take the exponents in the FI-scheme as follows: k

a(n)

+n-2 n = 1,2,...

k - 1

Then pn

P{Gn

a)

11

k

PIG+k-1(k) = 1}.

n+k a(1) + Therefore, for any n = 1, 2.... we have the equality

(L(1, k), ... , L(n, k)) _ (L(1) + k - 1, ... , L(n) + k - 1),

where the record times L(n) correspond to the F°-scheme with exponents a(n)

_

k+n-2

n = 1, 2, ... .

k-1

Since the study of the record times in the FO-scheme uses the arguments which

were used in the classical record model, we leave the proof of some additional relations to the reader. EXERCISE 25.1. Show that f o r any n = 1, 2, ... and 1 < m(1) < m(2) < . - . < m(n) we have

P{L(1) = 1, L(2) = m(2),..., L(n) = m(n)} =

l a(m(r)) S(m(n)) 1 2 S(m(r) - 1) nn

EXERCISE 25.2. Show that L(1), L(2),... form a Markov chain with

P{L(n) = j I L(n - 1) = i} = S(i) ( and

P{L(n) > j I L(n - 1) = i} =

1

S(.

1

1)

S(:), SW

)

S(j)

>i.

EXERCISE 25.3. Let

A(n) = EN(n) = -1 a(1) a(j) +....}. a(j) j and let

.Fn = a{L(1), L(2), ... , L(n)} denote the a-algebra generated by the record times L(1), L(2),. .., L(n). Assume that as n --.oo. Then the random variables V(n) = A(L(n)) - n,

n = 1,2,...,

form a martingale with respect to the sequence of o-algebras Fn, and

E A(L(n)) = n,

n = 1,2,....

2. RECORD TIMES AND RECORD VALUES

114

EXERCISE 25.4. Let

n +)1) -, 1 as n -+ oc. S(n Then for any fixed k = 2, 3, ... the random variables S(L(n + k - 1)) S(L(n + 1)) S(L(n)) ' S(L(n + k)) S(L(n + 1))' S(L(n + 2))' are asymptotically independent and S(L(n)) S(n) -+ oo and

lim P nco S(L(n + 1))

<x =x,

0 < x < 1.

We will restrict ourselves to the results stated above, though there are yet a number of limit theorems for record and inter-record times in the F°-scheme which generalize the corresponding results for the classical record model. Lecture 26. INDEPENDENCE OF RECORD INDICATORS AND MAXIMA

In the previous lecture we discussed some problems concerning independence

of the record indicators in the F°-scheme. The relationship between indicators S1, b, ... , S. and maxima M(n) = max{Xl, X2, ... , Xn} is also of interest. We have the following result.

THEOREM 26.1. In the F°-scheme the record indicators {1i 2i ... , Cn and the random variable M(n) = max{Xl, X2, ... , X,.} are independent for n = 1, 2,... . PROOF. Since the probability integral transformation does not affect the ordering of random variables and hence preserves the distributions of record indicators,

it is sufficient to prove the theorem for F(x) = x, 0 < x < 1. One can easily see that in this case

P{M(n) < x} = xs(n),

(26.1)

0 < x < 1,

and for any r and 1 0,

and exponents satisfying the conditions of Theorem 27.1. Then (27.4)

P{X(n) - n < xn1/2} -+ F(x),

n -p oo.

When we studied in Lecture 18 the structure of limit distributions for record values in the classical scheme, we also considered first the exponential distribution and obtained relation (18.1). Then using the probability integral transformation

we obtained the three types of limit distributions for X(n). Now, starting with relation (27.4), which coincides with (18.1), and using the same arguments we arrive at the same set of limit distributions in the Fa-scheme. Thus we have the following result.

2s, RECORDS [.'Oil. DEPENDENT RANDOM VARIABLES

II)

THEOREM 27.4. Let X1,X2,... form an F"-scheme with continuous d.f. F and exponents which satisfy, as n

oo, the conditions

pn -+ 0,

S(n) -' oc,

(n)))

and

(A

Then the set of all limiting d.f 's T(x) for suitably centered and normalized record times X (n) comprises the following three types: Ti(x) = 'D (x);

for x > 0, for x < 0;

14) (-Ylogx), T2 , 7

(x) =

7's ,7

(x)

t f

0

(D(-y log(-x)),

for x < 0,

1

for x > 0,

where -y > 0.

REMARK 27.5. It can be shown that Theorem 27.4 remains valid for an F'scheme with exponents a(1), a(2).... satisfying the following condition: there ex-

ists p,0 1 and any xl, x2, ... , xn.

REMARK 28.6. The right-hand side of (28.4) is a mixture of distributions of the n-variate vector (X1, X2i ... , Xn) with i.i.d. components having the common d.f. Fe. Assume additionally that d.f.'s Fe are continuous for almost all 0 (with respect to G). Then, as we know from Lecture 15, for any fixed 9

P{Xo(n) < x} =

1

(n

1)!

f

log(1-Fe(y))

so that (28.5)

P{X(n) < x} = f P{Xg(n) m < x} G(dO) 00

_00 (n

1

- 1)

1-Iog(l-Fe(x))

u"-1e

du,

2. RECORD TIMES AND RECORD VALUES

122

Note that in this case the joint distributions of record values X1, X2.... in an infinite sequence of symmetrically dependent random variables are also representable in the form

P{X(1) <xii...,X(n) <xn} =

r00

J

Fn.e(xi,...,xn)G(d9),

00

where

Fn,9(x1,... , xn) = P{X9(1) < xl,... , X9(n) < xn},

n = 1, 2, ... ,

are the d.f.'s of record values X9(1), X9(2), ... in a sequence of i.i.d. random variables with common d.f. F9. EXAMPLE 28.7. Consider a stationary Gaussian sequence X1, X2, ... with moments

O 2. i=1

For x > y and j = 1, 2, ... we have Q(j, x, y) = 0. PROOF. For any

1 = m(1) < m(2) < . . < m(n + l) and t1 m}=p"

M =0,1.....

Therefore n

n

m}=HP{X, >- m}=fipm=pm J=1

J=1

and

P{Y, = m} = P{Yn > m} - P{Yn > m + 1} = ; I - P)Pm, where p =

ra=1 Pi. 137

M = 0,1....,

Arr'ENDIX 2

188

Answer: the random variable min{Xi, X-2,..., X,) has the geometric distribution with parameter 71

P = f Pj. j=1

EXERCISE 2.4 (solution and answer). Let us find the probabilities P{Mn < x, Mn+1 < y}. Since Mn+1 > M,,, we have y} = P{Mn+1 < y} = Ff+l (y),

P{lbin < x,

ify<x. Fory>xwehave P{Mn, < x, Mn+1 < y} = P{Mn < x, X,,+1 < y} = Fn(r)F(y). Answer: P{Mn < x, Mn+l < y} = Fn (min(x, y)) F(y). EXERCISE 2.5 (solution). Using (2.1) we find that

P{-Xn_k+l,n < x} = P{Xn-k+l,n > -x} = 1 - P{Xn_k+1.n !5 -x}

=1-E

(fl)Fm(_X+O)(l_F(X+O))n_m

m=n- k + 1 m k

_

Fm(-x+0)(1

m=o Cn/ m

-

F(-x+0))n

m.

Since F(-x + 0) = 1 - F(x), 1 - F(-x + 0) = F(x), and (m) = (n nm), we obtain that

n-k

E m=U

(n) Fm(-x+0)(1 -

F(-x+0))n

M

_

F(r))n-m

(n)Fm(x)(1 -

m m=k

= P{Xk.n < x},

i.e., Xk,n and -Xn-k+1,n have the same distribution. Taking 2n + 1 instead of n and n + 1 instead of k, we see that d

Xn+1,2n+l = -Xn+1,2n+1,

which means that the sample median Xn+1,2n+i in this case has a symmetric distribution. EXERCISE 3.1 (solution). The uniform distribution U([0,11) is symmetric about the point 2. Hence by reasoning similar to that in Exercise 2.5 we obtain Ukn

k=1,...,n.

1-Un_k+l,ni

Now (3.7) and the probability integral transformation imply that

Wr= ( Ur,n \r d (1-Un-r+l,nlr d 1 - Un_r,n J `Ur+l,n J = exp { - r(Zn-r+1,n - Zn_r)}

d

(exp(-Zn-r+i.n)lr

I\

exp/(-Zn-r.n) J

exp{-vr}

d

1 - Ur

d

Ur,

HINTS, SOLUTIONS. AND ANSWERS

1319

i.e., the random variables W1, W2,. .. , W,, are independent and uniformly distributed on (0, 11.

EXERCISE 3.2 (solution). We know from Remark 3.7 that the joint density , vn.f 1), of the sums S1, ... , Sn, Sn+l in the domain 0 < vi < function, f (vi , < vn+I is given by v2 < , , .

f (vl,... vn+1) = exp(-vn+i ) ,

As is well known, the sum Sn+I has the gamma distribution with density

g(x) =

xn exp(-x) n!

Hence the conditional density of the first n sums given that Sn+1 = 1 equals

f('Ui,Vn,1) =n!

if 0 min{n-r+1,(n-s+1)/2}'

III

IIIN'I'S. SOLU'T'IONS. AND ANSWI-:Its

EXERCISE 6.3 (solution). We have the obvious equalities

X2,1 = max{XI, X2} = X1 + max{0, X2 - X1 } and

EX2,2=EX1+Emax{0,X2-XI}=al-rEmax{0,Y}. where Y = X2 - Xl has the normal distribution with mean

EY=a2-a,=aa and variance

a 2 = Var Y = Var X1 + Var X2 - 2 cov (X, , X2) = ai t a2 - 2pa I a2.

Let a > 0. Write Y = as + aV, where V is a random variable with standard normal distribution. Then EX2,2 = al

+aa+aE

max{-a, V}.

It is easily seen that 00

E max{-a, VI = -aP{V < -a} +

J

xV(x) dx = -a(b(-a) -

Jx d (x)

n

= -al,(-a) + w(-a) = -a(1 - 0(a)) + p(a). Hence we finally obtain that

E X2,2 = al + as + alp(a) - aa(1 - t(a)) = al + ap(a) + aa4D(a). If Var Y = 0, then

E X2,2 = a, + E max{O, Y} = al + E max{0, a2 - a, } = a, + max{0, a2 - a, } = max{a,, a2 l. EXERCISE 7.1 (solution). It follows from the equalities

cov(Xr,n-X)X=0 and that n

E COV(Xr,n, Xs.n) = COV(Xr,n, Xl,n + ,

+ Xn.n)

s=1 1.

EXERCISE 7.2 (solution). For the underlying random variables XI, X2, ... , Xn write the sum (min(Xr, (lnax(Xr, Xs))k

X,))m

1 m, Nl.,, = Xk} =

P{N(n) > m} n

Hence

P{N(n) > m I Mn = Xk} =

P{N(P{Mm=

Xk} Xk}

n

= PIN(n) > m}

n+m

i.e., for any k = 1, 2, ... , n the random variable N(n) and the event {M" = Xk } are independent. Furthermore, P{N(n) > rn I Mn = x} = P{ coax{X.n+1, ... , Xn+.n < x}} = Fm(x). Answers:

n = 1,2,..., m = 1,2,..., k = 1,...,n. EXERCISE 14.1 (solution). Let ao

H(z, s) _

00

P(L(n) = k}z"sk.

Qn(s)z" n=1k=n

n=I

Using the binomial expansion 00

(1 - s)m =

where

F(-1)"

(a) -

a(a - 1) ... (a - n + 1)

n

n=0

n!

we get by (14.11) and (14.12) 00

00

H(z, s) = 1: 1: j'P{N(k - 1) = n - 1}znsk n=1k=n

00 zskk-1P k=1

z z

00

k-1

n z"=z

k n=0

skz(z+1)...(z+k-2) k- 1.

k=1 k

z)(-z) I... (-z - k + 2) = z

1

k=1 z

z

c(-s)k C1 k z

1

k=1 z

((1 - s)1

(exp{(1 - z)log(1 - s)} - 1).

z - 1) =

z-1 z -1 It remains to expand the last expression in terms of z" to obtain Qn(s) = E sL(n) = I - (1 - s)

E

(- log(I - s))k 41

k=0

We have already mentioned that k-1-1

M=0

Xm

m!

e_x

1

(k - 1)!

J

°0

e-atk-1 dt,

0 < x < oc

l J

APPENDIX 2

146

(see, e.g., hint to Exercise 12.3). Hence

log(1 - s))k

x .

1

(n - 1)I

kI

k=o

-

I

e_tti-1 dt log( 1-a)

and

Iog(1-g)

1

t"-le-` dt.

Q" (S) = ( n - 1)!

EXERCISE 14.2 (solution). We restrict ourselves to the proof of asymptotic independence of the ratios Tt.n =

L(n + 1) L(n)

and

T2.n =

L(n + 2)

L(n+

1)

Taking into account the Markov property of the sequence of record times and the equality (14.3) and using some elements of the proof of Theorem 14.2 we obtain that for any u > 1 and v > 1 00

PIT,,. > u, T2,n > v} = E P{Tl,n > u, T2,n > v I L(n) = m}P{L(n) = m} m=n 00

_ E E P{L(n + 1) = r, L(n + 2) > yr I L(n) = m}P{L(n) = m} rn=nr>um 00

_ E E P{L(n + 2) > yr I L(n + 1) = r, L(n) = m} m=n r>um

x P{L(n + 1) = r I L(n) = m}P{L(n) = m} 00

= 1: 1: P{L(n + 2) > yr I L(n + 1) = r} m=n r>um 00

x P{L(n+ 1) = r I L(n) = m}P{L(n) = m}

E

(vr)P{L(n + 1) = r I L(n) = m}P{L(n) = m}

rn=n r>um

E00 P{L(n+ 1) > um I L(n) = m}P{L(n) = m} _ (v +O(n)) m=n = m} (v +0(n1)) n (urn] P {L (n )

-(1+0(1))(1+0(1)) EP{L(n)=m}= 1 +O(1 v

n

u

n

uv

m=n

so that

limP{Tl,n > u,T2,n > v} = 1 , uV

n - oo,

which means the asymptotic independence of the ratios T1,n and T2.n.

n

HINTS, SOLUTIONS, AND ANSWERS

147

EXERCISE 14.3 (solution). To avoid cumbersome calculations, we will prove,

as before, independence of two random variables T(n) and T(n + 1). For any integers u > 1 and v > 1 we have L(n)1)

P{T(n) > u, T(n + 1) > v} = P j L(

> u, L(L(n)1) > V

By the above arguments and using additionally the fact that u and v are integers we obtain L(n)

P L(n

L(n + 1)

L(n)

> u,

> v

- 1) 00

=

E

P{L(n + 2) > yr I L(n + 1) = r}

m=n r>um

x P{L(n + 1) = r I L(n) = m}P{L(n) = m} 00

VV

vrP{L(n + 1) = r I L(n) = m}P{L(n) = m}

m=rt r>um 00 1

V

E P{L(n + 1) > um I L(n) = m}P{L(n) = m} m=n 00

-

= 1 E M P{L(n) = m} = V

m=n um

1uv E P{L(n) = m} = uv 1

m=n

Thus we get the equality

P{T(n) > u,T(n + 1) > v} = v = P{T(n) > u}P{T(n + 1) > v}, which implies independence of T(n) and T(n + 1) as well as the fact that

P{T(n) = r} = P{T(n) > r-1}-P{T(n) > r} =

1

r11 r

r(r 1 1)'

r > 2.

EXERCISE 15.1 (hint). Use (15.7) and Remark 4.3.

EXERCISE 16.1 (hint). Express the probabilities

P{X(1) = il, X(2) = i2i ...,X(n) = in } in terms of the distributions of the indicators ?7k as in Theorem 16.1 and use the independence of these indicators. EXERCISE 16.2 (hint). Use (16.8) to find the probabilities

P{X(n+1)=jn+1 IX(n)=in,...,X(1)=J1} P{X(n+ 1) = j,,+1, X(n) = jn,... X(1) = j1}

P{X(n)=jn,. ,X(1)=j1} and compare the expression thus obtained with (16.6).

APPENDIX 2

118

EXERCISE 16.3 (solution and answer). It follows from Theorem 16.8 and Representation 16.9 that

P{X,,,(n) > 'm} = P{/10 + µ, +

+ Lm < n},

where po, pt, ... are i.i.d. random variables, since in our case

P{µ,,=m}=(1-Pn)Pn for any n, where

P

P{X = n}

-

=(1-P),

P{X >n}

n=0,1,.

.

This implies that the sum µ, + AI + + µ,,, has the negative binomial distribution with parameters (1 - p) and (m + 1). Consequently,

P{X.(n) < nil = P{µ + A, +

+ µ,n > n}

(r+m)(l_p)rpm+l r>n

r

and, in particular, P{X.(n)=0}=E(1-p)rp=(1-p)n.

r>n

Answer: For any n = 1, 2.... and m = 1, 2, .. .

P{X.(n) < m} = pn` 1 (r+rn_ 11(1 - p)r. r>n J EXERCISE 18.1 (answer). With each d.f. F associate the d.f.

F(x) = 1 - exp {- V1-_Iog(1 -- F(x))} I.

Let N(n) = [exp(n)'12]. The following result is due to Resnick (1973a): F belongs to the domain of attraction of the limiting distribution H for maxima if and only if F belongs to the domain of attraction of the limiting distribution

G(x) = fi( - log(- logH(x))) for record values. The centering and normalizing constants for maxima (a(n) and b(n)) and for records (A(n) and B(n)) are related by

A(n) = a(N(n)),

B(n) = b(N(n)).

EXERCISE 18.2 (hint). Combining the result of Exercise 18.1 and Examples 10.3 and 10.4 with d.f.'s

PI(x)=1-x--', x>1, and F2(x)=1-(1-x)', 0<x