Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability: Volume 3 Contributions to Probability Theory [Reprint 2020 ed.] 9780520375918

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PROCEEDINGS SIXTH

BERKELEY

OF

THE

SYMPOSIUM

VOLUME III

PROCEEDINGS of the SIXTH BERKELEY SYMPOSIUM ON MATHEMATICAL STATISTICS AND PROBABILITY Held at the Statistical Laboratory University of California June 21-July 18, 1970 with the support of

University of California National Science Foundation Air Force Office of Scientific Research Army Research Office Office of Naval Research

V O L U M E

III

PROBABILITY T H E O R Y

EDITED BY L U C I E N M . L E C A M , J E R Z Y N E Y M A N , AND E L I Z A B E T H L. S C O T T

UNIVERSITY OF CALIFORNIA PRESS BERKELEY AND LOS ANGELES

1972

UNIVERSITY OF CALIFORNIA PRESS B E R K E L E Y AND LOS ANGELES CALIFORNIA

CAMBRIDGE UNIVERSITY PRESS LONDON, ENGLAND

COPYRIGHT ©

1 9 7 2 , BY

T H E REGENTS O F T H E UNIVERSITY O F CALIFORNIA

The United States Government and its offices, agents, and employees, acting within the scope of their duties, may reproduce, publish, and use this material in whole or in part for governmental purposes without payment of royalties thereon or therefor. The publication or republication by the government either separately or in a public document of any material in which copyright subsists shall not be taken to cause any abridgment or annulment of the copyright or to authorize any use or appropriation of such copyright material without the consent of the copyright proprietor. ISBN: 0-520-02185-1 LIBRARY O F CONGRESS CATALOG CARD N U M B E R :

49-8189

P R I N T E D IN T H E U N I T E D STATES O F AMERICA

CONTENTS OF PROCEEDINGS VOLUMES I, II, AND III Volume I—Theory of Statistics

General Theory R . J. B E R A N , Upper and lower risks and minimax procedures. C. R . B L Y T H and D. M. R O B E R T S , On inequalities of Cramer-Rao type and admissibility proofs. J . O O S T E R H O P F and W. R . VAN ZWET, The likelihood ratio test for the multinomial distribution. W. E. S T R A W D E R M A N , On the existence of proper Bayes minimax estimators of the mean of a multivariate normal distribution.

Sequential Analysis P. J . B I C K E L and J . Y A H A V , On the Wiener process approximation to Bayesian Sequential testing problems. Y u . V. L I N N I R and I. V. R O M A N O V S R Y . Some new results in sequential estimation theory. R. M I L L E R , Sequential rank tests—one sample case. R . A. W I J S M A N , Examples of exponentially bounded stopping time of invariant sequential probability ratio tests when the model may be false.

Asymptotic Theory R . R. B A H A D U R and M. R A G H A V A C H A R I , Some asymptotic properties of likelihood ratios on general sample spaces. D. M. CHIBISOV, On the normal approximation for a certain class of statistics. J . H A J E K , Local asymptotic minimax and admissibility in estimation. R. A. JOHNSON and G. G. R O U S S A S , Applications of contiguity to multiparameter hypotheses testing. J. K I E F E R , Iterated logarithm analogues for sample quantiles when p„|0. L. L E CAM, Limits of experiments. M. D. P E R L M A N , On the strong consistency of approximate maximum likelihood estimators. P. SWITZER, Efficiency robustness of estimators.

Nonparametric Procedures R. E. B A R L O W and K . A. D O R S U M , Isotonic tests for convex orderings. Z. W. B I R N B A U M , Asymptotically distribution free statistics similar to student's t. K . A. D O R S U M , Decision theory for some nonparametric models. J . M. H A M M E R S L E Y , A few seedlings of research. A. W. M A R S H A L L and F. PROSCHAN, Classes of distributions applicable in replacement with renewal theory implications. R . P Y R E , Spacings revisited. H. R U B I N , On large sample properties of certain nonparametric procedures. I. R. SAVAGE and J. S E T H U R A M A N , Asymptotic distribution of the log likelihood ratio based on ranks in the two sample problem. I. VINCZE, On some results and problems in connection with statistics of the Kolmogorov-Smirnov type.

Regression Analysis T. W. A N D E R S O N , Efficient estimation of regression coefficients in time series. D. R B R I L L I N G E R . The spectral analysis of stationary interval functions. H. B U H L M A N N , Credibility procedures. W. G. COCHRAN, Some effects of errors of measurement on linear regression. L. J . G L E S E R and I. O L R I N , Estimation for a regression model with an unknown covariance matrix. J . M. HOEM, On the statistical theory of analytic graduation. v

vi

C O N T E N T S OF P R O C E E D I N G S

Multivariate Analysis C. R. R A O and S. K. MITRA, Generalized inverse of a matrix and its applications. H. C H E R N O F F , Metric considerations in cluster analysis. F. N. DAVID, Measurement of diversity. L. A. G O O D M A N , Some multiplicative models for the analysis of cross classified data. T. O R C H A R D and M. A. W O O D B U R Y , A missing information principle: theory and applications. M. SOBEL and G. H. WEISS, Recent results on using the play-the-winner sampling rule with binomial selection problems. M. ZELEN, Exact significance tests for contingency tables embedded in a 2" classification.

Volume II—Probability Theory Introduction J. L. DOOB, William Feller 1906-1970. M. KAC, William Feller, in memoriam. L. K. SCHMETTERER, Alfred Renyi, in memoriam.

Measure Theory D. W. M Ü L L E R , Randomness and extrapolation. R. M. B L U M E N T H A L and H. H. CORSON, On continuous collections of measures. G. DEBREU and D. SCHMEIDLER, The Radon-Nikodym derivative of a correspondence. R. DUDLEY, A counterexample on measurable processes. Z. F R O L i K , Projective limits of measure spaces. W. HILDENBRAND, Metric measure spaces of economic agents. C. IONESCU TULCEA, Liftings commuting with translations. J. H. B. K E M P E R M A N , On a class of moment problems. D. M AH AR AM, Consistent extensions of linear functionals and of probability measures. H. ROSENTHAL, On the span in 1 / of sequences of independent random variables. L. K. SCHMETTERER, On Poisson laws and related questions. M. L. STRAF, Weak convergence of stochastic processes with several parameters.

Inequalities D. L. B U R K H O L D E R , B. J. DAVIS, and R. F. G U N D Y , Integral inequalities for convex functions of operators on martingales. S. DAS GUPTA, M. L. EATON, I. OLKIN, M. P E R L M A N , L. J. SAVAGE, and M. SOBEL, Inequalities on the probability content of convex regions for elliptically contoured distributions.

Combinatorial Analysis P. DOUBILET, G.-C. ROTA, and R. STANLEY, On the foundations of combinatorial theory, VI: the idea of generating function.

Ergodic Theory S. K A K U T A N I , Strictly ergodic symbolic dynamical systems. W. K R I E G E R . On unique ergodicity. D. S. ORNSTEIN, On the root problem in ergodic theory.

Gaussian Processes J. F E L D M A N , Sets of boundedness and continuity for the canonical normal process. A. M. GARSIA, Continuity properties of Gaussian processes with multidimensional time parameter. G. K A L L I A N P U R and M. N A D K A R N I , Supports of Gaussian measures. R. S. LIPTSER and A. M. SHIRYAYEV, Statistics of conditionally Gaussian random sequences. M. B. M A R C U S and L. A. SHEPP, Sample behavior of Gaussian processes. S. OREY, Growth rate of certain Gaussian processes.

C O N T E N T S OF P R O C E E D I N G S

vii

Central Limit Theorem R. N. BHATTACHARYA, Recent results on refinements of the central limit theorem. R. F. C O G B U R N , The central limit theorem for Markov processes. A. DVORETZKY, Asymptotic normality of sums of dependent random variables. B. V. G N E D E N K O , Limit theorems for sums of a random number of positive independent random variables. M. ROSENBLATT, Central limit theorem for stationary processes. V. V. SAZONOV, On a bound for the rate of convergence in the multidimensional central limit theorem. C. STEIN, A bound for the error in the normal approximation to the distribution of a sum of dependent random variables.

Volume III—Probability Theory Passage Problems Yu. K. BELYAYEV, Point processes and first passage problems. A. A. BOROVKOV, Limit theorems for random walks with boundaries. N. C. JAIN and W. E. PRUITT, The range of random walk. H. ROBBINS and D. S I E G M U N D , On the law of the iterated logarithm for maxima and minima. A. D. SOLOVIEV, Asymptotic distribution of the moment of first crossing of a high level by a birth and death process.

Markov Processes—Potential Theory R. G. AZENCOTT and P. C ARTIER, Martin boundaries of random walks on locally compact groups. J. L. DOOB, The structure of a Markov chain. S. P O R T and C. STONE, Classical potential theory and Brownian motion. S. P O R T and C. STONE, Logarithmic potentials and planar Brownian motion. K. SATO, Potential operators for Markov processes.

Markov Processes— Trajectories—Functionals R. GETOOR, Approximations of continuous additive functionals. K. ITO, Poisson point processes attached to Markov processes. J. F. C. K I N G M A N , Regenerative phenomena and the characterization of Markov transition probabilities. E. J. McSH ANE, Stochastic differential equations and models of random processes. P. A. MEYER, R. SMYTHE, and J. WALSH, Birth and death of Markov processes. P. W. MILLAR, Stochastic integrals and processes with stationary independent increments. D. W. STROOCK and S. R. S. V A R A D H A N , On the support of diffusion processes with applications to the strong maximum principle. D. W. STROOCK and S. R. S. V A R A D H A N , Diffusion processes.

Point Processes, Branching Processes R. V. AMBARTSUMIAN, On random fields of segments and random mosaic on a plane. H. SOLOMON and P. C. C. W A N G , Nonhomogeneous Poisson fields of random lines with applications to traffic flow. D. R. COX and P. A. W. LEWIS, Multivariate point processes. M. R. LEADBETTER, On basic results of point process theory. W. J. BUHLER, The distribution of generations and other aspects of the family structure of branching processes. P. S. PURI, A method for studying the integral functionals of stochastic processes with applications: III. W. A. O'N. W A U G H , Uses of the sojourn time series for the Markovian birth process. J. GANI, First emptiness problems in queueing, storage, and traffic theory. H. E. DANIELS, Kuhn-Grun type approximations for polymer chain distributions. L. K A T Z and M. SOBEL, Coverage of generalized chess boards by randomly placed rooks. R. HOLLEY, Pressure and Helmholtz free energy in a dynamic model of a lattice gas. D. MOLLISON, The rate of spatial propagation of simple epidemics. W. H. OLSON and V. R. R. UPPULURI, Asymptotic distribution of eigenvalues or random matrices.

viii

CONTENTS OF PROCEEDINGS

Information and Control R . S. B U C Y , A priori b o u n d s f o r t h e Riccati e q u a t i o n . T . F E R G U S O N , Lose a dollar o r d o u b l e y o u r f o r t u n e . H . J. K U S H N E R , Necessary c o n d i t i o n s f o r discrete p a r a m e t e r stochastic o p t i m i z a t i o n p r o b l e m s . P. V A R A I Y A , Differential games. E. C. P O S N E R a n d E. R . R O D E M I C H , Epsilon e n t r o p y of p r o b a b i l i t y distributions.

PREFACE S Y M P O S I A ON M A T H E M A T I C A L S T A T I S T I C S A N D P R O B A B I L I T Y have been held at five year intervals since 1945. The Sixth Berkeley Symposium was divided into four sessions. The first took place from June 21 to July 18, 1970. It covered mostly topics in statistical theory and in theoretical and applied probability. The second session was held from April 9 to April 12, 1971 on the special subject of evolution with emphasis on studies of evolution conducted at the molecular level. The third session held in June 1971 was devoted to problems of biology and health. A fourth session on pollution was held in July 1971. The first three volumes of the Proceedings cover papers presented in June and July, 1970, as well as papers which were sent to us at that time, but could not be presented in person by their authors. The first volume is entirely devoted to statistics. The second and third are devoted to contributions in probability. Allocation of the papers to the three volumes was made in a manner which we hope is fairly rational, but with an unavoidable amount of arbitrariness and randomness. In the event of doubt, a general index should help the prospective reader locate the desired contribution. The Berkeley Symposia differ substantially from most other scientific meetings in that they are intended to provide an extended period of contact between participants from all countries in the world. In addition, an effort is made to promote cross contacts between scholars whose fields of specialization cover a broad spectrum from pure probability to applied statistics. However, these fields have expanded so rapidly in the past decades that it is no longer possible to touch upon every domain in a few weeks only. Since time limits the number of invited lectures, the selection of speakers is becoming rapidly an impossible task. We could only sample the abundance of available talent. For this selection, as well as for several other important matters, we were privileged to have the assistance of an advisory committee consisting of Professors Z. W. Birnbaum and L. Schmetterer, representatives of the Institute of Mathematical Statistics, and of Professor Steven Orey, delegate of the American Mathematical Society. The visible success of our gathering is in no small measure attributable to the help we have received from this committee and other scientific friends. BERKELEY

A conference which extends over six weeks with participants from various parts of the world entails expenses. In this respect we feel fortunate that in spite of the general shortage of funds, the University of California and the Federal Agencies found it possible to support our enterprise. We are grateful for the allocation of funds from the Russell S. Springer Memorial Foundation, the National Science Foundation, the Office of Naval Research, the Army Research Office, the Air Force Office of Scientific Research, and the National Institutes of Health, which contributed particularly to the sessions on evolution and on problems of biology and health. In addition the pollution session received support from the Atomic Energy Commission. The organization of the meetings fell under the responsibility of the underIX

X

PREFACE

signed with the able help of the staff of the Statistical Laboratory and of the Department of Statistics. For assistance with travel arrangements and various organizational matters, special thanks are due to Mrs. Barbara Gaugl. The end of the actual meeting signals the end of a very exciting period, but not the end of our task, since the editing and publishing of over 3,000 pages of typewritten material still requires an expenditure of time and effort. In this respect we are indebted to Dr. Morris Friedman for translations of Russian manuscripts. We are particularly grateful to Dr. Amiel Feinstein and Mrs. Margaret Stein who not only translated such manuscripts but acted as editors, checking the references and even verifying the accuracy of mathematical results. The actual editing and marking of manuscripts was not easy since we attempted to follow a uniform style. We benefitted from the talent and skill of Mrs. Virginia Thompson who also assumed responsibility for organizing and supervising the assistant editors, Miss Carol Conti, Mrs. Margaret Darland, and Miss Jean Kettler. We are extremely grateful to all the editors for the knowledge and patience they have devoted to these manuscripts. In the actual publication of the material the University of California Press maintained their tradition of excellence. The typesetting was performed by the staff of Oliver Burridge Filmsetting Ltd., in Sussex, England. The meetings of the Sixth Symposium were saddened by the absence of two of our long time friends and regular participants, William Feller and Alfred Renyi. Professors J. L. Doob and M a r k Kac were kind enough to write a short appreciation of Feller. For a similar appreciation of Renyi, we are indebted to Professor L. Schmetterer. The texts appear at the beginning of the second volume. L.LC.

J.N.

E.L.S.

CONTENTS Passage

Problems

Yu. K. BELYAYEV—Point Processes and First Passage Problems

1

A. A. BOROVKOV—Limit Theorems for Random Walks with Boundaries

19

N. C.

and W. E. PRUITT—The Range of Random Walk

31

R O B B I N S and D . S I E G M U N D — O n the Law of the Iterated Logarithm for Maxima and Minima

51

A. D. SOLOVIEV—Asymptotic Distribution of the Moment of First Crossing of a High Level by a Birth and Death Process

71

Markov Processes—Potential Theory R. G. A Z E N C O T T and P. CARTIER—Martin Boundaries of Random Walks on Locally Compact Groups

87

H.

JAIN

J. L. DOOB—The Structure of a Markov Chain S.

S.

131

P O R T and C. STONE—Classical Potential Theory Brownian Motion

and 143

P O R T and C . STONE—Logarithmic Potentials and Planar Brownian Motion

K. SATO—Potential Operators for Markov Processes

.

193

Markov Processes—Trajectories—Functionals R. GETOOR—Approximation of Continuous Additive Functionals

213

K. ITO—Poisson Point Processes Attached to Markov Processes

225

J. F. C. KINGMAN—Regenerative Phenomena and the Characterization of Markov Transition Probabilities

241

E. J. MCSHANE—Stochastic Differential Equations and Models of Random Processes

263

P.

A . M E Y E R , R . SMYTHE,

and

J.

.

.

177

WALSH—Birth and Death of

Markov Processes

295 xi

xii

CONTENTS

P.

W. M I L L A R — Stochastic Integrals Stationary Independent Increments

and

Processes

with 307

D.

W . STROOCK: and S . R . S . VARADHAN—On the Support of Diffusion Processes with Applications to the Strong Maximum Principle

D

W . STROOCK

and

S. R . S.

VARADHAN—Diffusion Processes

Point Processes, Branching Processes R. V. AMBARTSUMIAN—Random Fields of Segments R a n d o m Mosaic on a Plane H.

333 361

and 369

SOLOMON and P. C . C . WANG—Nonhomogeneous Poisson Fields of R a n d o m Lines with Applications to Traffic Flow .

383

D. R. Cox and P. A. W. LEWIS—Multivariate Point Processes .

401

R . L E A D B E T T E R — O n Basic Results of Point Process Theory

449

M.

W. J. BUHLER—The Distribution of Generations and Other Aspects of the Family Structure of Branching Processes . .

463 •

P. S. PURI—A Method for Studying the Integral Functionals of Stochastic Processes with Applications, III

481

W. A. O ' N . WAUGH—Uses of the Sojourn Time Series for the Markovian Birth Process

501

J. GANI—First Emptiness Problems in Queueing, Storage, and Traffic Theory

515

H. E. DANIELS—Kuhn-GriinType Approximations for Polymer Chain Distributions

533

L.

K A T Z and M. SOBEL—Coverage of Generalized Chess Boards by Randomly Placed R o o k s

555

R. HOLLEY—Pressure and Helmholtz Free Energy in a Dynamic Model of a Lattice Gas

565

D. MOLLISON—The R a t e of Spatial Propagation of Simple Epidemics

579

H. O L S O N and V . R . R. UPPULURI—Asymptotic Distribution of Eigenvalues of R a n d o m Matrices

615

W.

CONTENTS

xiii

Information and Control R. S. B U C Y — A Priori Bounds for the Riccati Equation . T. FERGUSON—Lose a Dollar or Double Your Fortune .

645 .

657

H. J. KUSHNER—Necessary Conditions for Discrete Parameter Stochastic Optimization Problems

667

P. VARAIYA—Differential G a m e s

687

E.

.

C . P O S N E R and E. R . RODEMICH—Epsilon Entropy of Probability Distributions

699

PAUL LÉVY 1886-1971 At a time when the composition of the present volume was already far advanced, we heard, with sorrow, that Paul Levy died on December 15, 1971. The printing schedule did not permit us to present here a memorial article appropriate to the unique stature of Levy in the probabilistic field. The following text offers only a few indications on Levy's career and his work. A more extensive article, with a bibliography, is expected to appear in the first issue of the Annals of Probability. We are greatly indebted to Mrs. Levy and Professor Laurent Schwartz for communicating to us documents on Paul Levy's career. Ed.

Paul Levy was born in Paris on September 15, 1886, in a family with a strong traditional link with mathematics. Both his father, Lucien Lévy, and his grandfather were professors of mathematics. In 1904, Paul Levy secured first place at the competitive examination for entrance to the Ecole Normale Supérieure. He also placed second in the Ecole Polytechnique examination. For various reasons, Levy elected to join the École Polytechnique, which then enjoyed the majestic presence of Henri Poincaré. This was a surprising choice for someone interested in mathematical research, but it may by chance have saved Levy's life by placing him in a less vulnerable position during the first world war. After leaving Polytechnique, Lévy became an engineering student at the Ecole des Mines (Paris) and finally, in 1910, Professor at the Ecole des Mines of Saint Etienne. Paul Levy's doctoral thesis was accepted in 1911 by a jury consisting of Poincaré, Hadamard, and E. Picard. He joined the Ecole Polytechnique in 1913 and replaced G. Humbert as Professeur d'Analyse in 1920. Lévy kept this position until 1959. In spite of poor health, Lévy continued an uninterrupted stream of mathematical activity almost up to the time of his death on December 15, 1971. This activity resulted in the publication of about 260 scientific papers. In addition, Lévy published four scientific monographs as follows : (i) Leçons d'Analyse Fonctionnelle, Gauthier Villars, Paris, 1922 (439 pages), 2nd edition, 1951, with the title, Problèmes Concrets d'Analyse Fonctionnelle. (ii) Calcul des Probabilités, Gauthier Villars, Paris, 1925 (350 pages). (iii) Théorie de l'Addition des Variables Aléatoires, Gauthier Villars, Paris, 1937 (17 + 328 pages), 2nd edition, 1954. (iv) Processus Stochastiques et Mouvement Brownien, Gauthier Villars, Paris, 1948 (438 pages), 2nd edition, 1965. Paul Lévy also published in 1935 and in 1964 Notices on his scientific work and in 1970 an autobiographical volume, Quelques Aspects de la Pensée d'un Mathématicien (Paris, A. Blanchard, 1970, 222 pages). This last book gives a xv

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charming and very candid description of Levy's mathematical thoughts from his independent discovery in 1902 of von Koch's tangentless continuous curve up to his hesitant philosophical speculations on Paul Cohen's proof of the indecidability of the continuum hypothesis. Paul Levy's own work has had immense influence especially on the developments in the field of probability and stochastic processes. His magnificent Théorie de l'Addition des Variables Aléatoires was published 35 years ago. Since then the field of probability has undergone a period of unbridled expansion. In spite of this the reader who will take the trouble to ponder Levy's writings of the thirties will still find them permeated with unbelievable and yet unexhausted treasures. These writings, in beautiful fluent classical French, do not have the dry formal structure younger mathematicians have been trained to expect. Levy, with his powerful intuition, seemed to be able to "see" the sample functions of stochastic processes or the fluctuations in a sequence of random variables. They were intimate friends which he described, pausing only from time to time to state a more formal proposition. This does not make his proofs any less rigorous than the more usual ones, but it may prevent the hurried reader from assimilating the richness of the thought and even prompted Fréchet to comment that "Your results are more or less complex according to one's own perspective." A description of the specifics of Levy's work would be worth a volume itself. We shall give here only a few comments on some of the highlights. Paul Levy's early work, represented for instance by his doctoral thesis in 1911 and culminating in the 1922 volume on Analyse Fonctionnelle, revolves around the extension to an infinity of dimensions of the classical theorems relative to first and second order partial differential equations. This was influenced by Volterra's style of study of "functions of lines" and Hadamard's questions concerning the manner in which Green's functions depend on the contours of the domain. Paul Levy was able to extend many of the results relative to first order partial differential equations, but the second order equations led to problems of a very different nature. The study of the Laplacian in Hilbert space led both Gâteaux and Levy to introduce independently the idea of mean values taken on balls or other convex subsets of Hilbert space. Gâteaux, who was killed at the beginning of World War I, left unpublished manuscripts. Hadamard gave Lévy the task of rewriting them for publication. Some of the unsolved questions raised by these papers were combined with Levy's own ideas and at least partially solved in the 1922 volume. This work of Levy is not well known today, perhaps because functional analysis took around that time a rather more abstract direction with the introduction of Banach spaces and the ensuing emphasis on the general theory of linear operations, and perhaps because the extension of the Laplacian to Hilbert space by Levy and Gâteaux turned out to be different from that needed in quantum field theory. The rather deep aspects of the geometry of Hilbert space studied by Levy still remain to be fully explored. One can mention, as part of the incidental contributions of Levy, his 1919 description of Lebesgue measure on the infinite dimensional cube carried out without knowledge of

PAUL LÉVY

xvii

Fréchet's 1915 paper on general cr-additive set functions and independently of the concurrent work of P. J. Danieli. In 1919 Levy was requested to give three lectures on the Calculus of Probabilities. This incident was to change radically the direction of Levy's work and the field of probability itself. The general shape of the field at that time was not particularly resplendent, and the texts available to Levy were even worse, ignoring in particular the works of Chebyshev and Liapounov. In this state of affairs, Levy set out to prove the proposition suggested by Laplace and Poincaré that an error which is a sum of many independent terms will have a distribution close to the Gaussian, unless the maximum term yields a substantial contribution. With the exception of a conjecture which was to be proved by Cramér in 1936, Levy achieved his stated goal in 1934. As a first step, Levy had to rethink what was meant by random variables and their distributions. For the description of these latter he introduces, independently of von Mises, the idea of cumulative distribution function and also an essentially measure-theoretic description relying on ordered countable families of finite partitions. At about the same time Levy starts using, under the name of characteristic function, the Fourier transform of a probability measure, gives an inversion formula, and proves that the relation between Fourier transform and probability measure is bicontinuous. This theorem, now very familiar, is better appreciated if one remembers that it does not extend to signed measures, that the correspondence is not at all uniformly continuous, and that the only inversion formulas available at that time were encumbered by various restrictions. With this tool, Levy proceeds to prove various versions of the central limit theorem and a number of propositions relative to symmetric or asymmetric stable laws. Some of the results including refinements of the previous work of G. Pólya and a version of Lindeberg's proof of the central limit theorem, can be found in Levy's Calcul des Probabilités of 1925. The book also contains a chapter on applications to the kinetic theory of gases and even some discussion of the use of trimmed means for estimation when the errors are not in the domain of attraction of the Gaussian distribution. It is curious to note that in 1925 Levy discusses attraction to stable distributions, but considers convergence to the Gaussian distribution exclusively for sums of variables which possess second moments. This restriction is notably absent ten years later in Levy's work and in Feller's work. After 1925 Levy continues to elaborate on the behavior of sums of independent variables, rediscovering and improving in particular Khinchin's law of the iterated logarithm. However, the most spectacular contributions to the subject can be found in three papers: the 1931 paper in Studia Mathematica, the voluminous paper in Journal de Mathématiques, Vol. 14 (1935), and the two part paper in the Bulletin des Sciences Mathématiques, Vol. 59 (1935). The method, used by Levy in 1931, of bounding the oscillations of successive sums of independent random variables by the dispersion of the last term and through symmetrization remains one of the most powerful available in this

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domain. In the independent case, the 1935 papers give necessary and sufficient conditions for a sum of variables to have a distribution close to the Gaussian under the assumption that individually the terms are small. An equivalent result was obtained independently and published at approximately the same date by W. Feller. However, Levy's description of the problem goes much deeper. He attempts to see what happens if the summands are not individually small and on this occasion formulates and elaborates the consequences of the conjecture that if X + Y is Gaussian so are X and Y. Cramer proved the validity of the conjecture in 1936, thereby clearing up the situation and prompting Levy to write his memorable monograph of 1937. This seemed to complete the program Levy had set for himself in 1919. However, in the meantime, Levy had enlarged the scope of the investigation to the case of dependent variables. The main results available in this domain were those of S. Bernstein (1927). Levy attacked the problem with such a flood of new ideas that even today the dust has not settled and it is not unusual to find papers which reprove particular cases of Levy's results of 1935. It is in these 1935 papers that the convergence theorems for series of independent variables are extended to the martingale case and that the central limit theorem is extended to martingales. For the latter, Levy considers martingales differences and then conditional variances. He measures "time" according to the sums of these conditional variances and first proves a theorem concerning martingales stopped at fixed "time." He then proceeds to show that the result remains valid for other stopping rules, provided that the variables so obtained do not differ too much from the fixed "time" sums. For a very recent result in this direction, the reader may consult a paper by A. Dvoretzky in the present Proceedings (Volume II). (The names "martingale" and "stopping times" do not occur in these papers. Doob, to whom many of the fundamental results on martingales are due, borrowed that name from J. Ville's analysis of gambling systems.) A glance at any modern text on probability or any of the standard journals will convince the reader that martingales have now penetrated the bulk of new developments. Since Paul Levy had first relied on characteristic functions for the proof of limit theorems, it is worth mentioning that curiously enough the 1937 volume makes very little use of this tool in proving the usual limit theorems. The only place where the Fourier transform appears essentially is in the Cramer-Levy theorem. Paul Levy mentioned several times that this was one of the very few instances where he could not obtain a proof by following his intuition. The fact that a nonvanishing entire characteristic function of order two must be Gaussian carries little probabilistic flavor and at the present time no intuitive proof exists. On the contrary, Levy's proof of the general central limit theorem retains a lot of intuitive appeal. To make it rigorous, Levy had first shown that the space of probability measures on the line may be metrized in such a way that if two variables X and Y are such that Pr {|X — Y| > e} ^ e, then they have distributions differing by at most £. Prohorov extended this result to separable metric

PAUL LÉVY

xix

spaces in 1953 and Strassen showed in 1965 that the property Levy requested of his distance can in fact be taken as a definition of Prohorov's distance. When looking at sums of independent variables Levy separates the values they take into a set of small values and a set of large values. Through the use of concentration inequalities, he shows that it is legitimate to treat the two sets as if they were independent. Kolmogorov revived and improved this method around 1955. He and his school have shown that it leads to many deep approximation results. In 1938, prompted by a question of J. Marcinkiewicz, Levy returns to the study of the Brownian motion process and publishes two fundamental memoirs on the subject in Compositio Mathematica and in the American Journal of Mathematics. In these memoirs, one finds the description of the sets of zeros of ordinary Brownian motion, the distributions of first passage times, and their relation to the increasing stable process of exponent 1/2, the arcsine law for percentage of time spent above a level, and a number of other related results including the first definitions and theorems on what is now called the "local time" of the process. The Compositio memoir deals only with one dimensional processes. The American Journal of Mathematics memoir considers a particle undergoing Brownian motion in the plane. There, Levy introduces stochastic integrals to give relations satisfied by the area between the Brownian curve and its chord, and gives results on the measure of the Brownian curve. Many of these results appear in his book of 1948. Levy's ideas on "local times" and on stochastic integrals were soon noticed by K . Ito whose book with H. P. McKean ( D i f f u s i o n Processes and Their Sample Paths, Springer-Verlag, Berlin, 1965) can be consulted for more recent development. Levy obtained in the early fifties several other groups of results on the measure of n dimensional Brownian motion and on the canonical representations of Laplacian random functions. Part of this, together with a mention of the beautiful results of Dvoretzky, Erdos, and Kakutani, was added to the 1965 edition of the book. The preceding memoirs refer to stochastic processes which may be multidimensional, or even Hilbert valued, but where the indexing set T is the real line. Around 1945, Levy starts the study of a very different object. Now the variables are real valued, but the index set T becomes an n dimensional space, or a sphere, or a Hilbert space. The underlying space T has then a metric structure and Levy requires that the expectation E\X(s) — X(t)\2 be the distance between s and t. The work described in the 1948 volume concerns mostly continuity properties of such a Gaussian process, including iterated logarithm laws. By 1955 Levy undertakes a deeper study of the processes and of their averages on spheres of varying radius and shows in particular that if T is infinite dimensional Hilbert space, the process is already determined by its values in arbitrarily small balls. The 1965 edition of Processus Stochastiques gives a summary of the main results obtained by Levy between 1955 and 1963, including a modification

XX

S I X T H B E R K E L E Y S Y M P O S I U M : L E CAM

of an integral representation formula of Chentsov and mention of improvements on Levy's results due to T. Hida and H. P. McKean. Another important part of Levy's work started in 1950 as a result of a conversation with K. L. Chung. The 1951 memoir in the Annales de l'Ecole Normale Supérieure gives a classification of the states of Markov processes with a countable state space, together with constructions of various Markov processes exhibiting all sorts of curious and seemingly pathological behavior. It is a peculiar trait of Levy's psychology that since his mind would follow the development in time of the trajectories, he could not conceive that one would dare call "Markov process" a process which was not strongly Markovian. This should be kept in mind while reading this memoir, which in spite of difficulties pointed out by K. L. Chung still gives a wonderful intuitive view of the situation. The above does not exhaust Levy's contributions to Probability, nor does it give justice to Levy's mathematical work in other directions. Paul Levy seems to have been the first to represent a stochastic process by a curve in Hilbert space. He also initiated the study of composition of random variables under laws other than addition and was led in this way to look at diffusion processes on a circle or a sphere. In pure analysis his early work includes several notes on the theory of functions of real or complex variables. Perhaps one of his most striking contributions to analysis is the famous theorem, often called theorem of Wiener-Lévy, according to which if a Fourier series converges absolutely to, say/, and if ---,it)v(di1)---v(dit),

4

SIXTH B E R K E L E Y S Y M P O S I U M :

BELYAYEV

where Xk(t1, • • • , tk) is called the A:-parameter function of the r a n d o m stream or random point set with respect to the measure v. Thus, with every r a n d o m point set S on \_T, .JiT~\ we can, generally speaking, associate the infinite sequence of A:-parametric measures, which in the presence of a fundamental system enables us to find the moments of higher order. One of the important unsolved problems is whether or not the sequence of parametric measures Afc (Ak) determines all probabilistic characteristics of S. Under very general conditions on r a n d o m streams which are homogeneous with respect to a group of motions in the space T, the parametric measure is proportional to the corresponding H a a r measure, which generalizes the wellknown result of Hincin [9] on the existence of the parameter for a stationary random stream on T = R1 (see Belyayev [4]). 2. Special systems of conditional probabilities In a large number of applied problems, it becomes necessary t o calculate the conditional probabilities of various events, when the conditioning event is the appearance in a r a n d o m point set of a point with given coordinates. T h a t this problem is not entirely simple is attested t o by the fact that a number of papers have been devoted t o the question of methods f o r determining the distribution function of the length of the interval between successive events f r o m a stationary stream on the line (McFadden [15], Ryll-Nardzewski [20], Matthes [14], Cramer and Leadbetter [7], and others). We shall assume that the r a n d o m point set S considered below is defined on \T, Jt T ~\, which we shall assume has a f u n d a m e n t a l system c6, and that the parametric measure is given by -1(A) = JA X(t)v(dt), where v(A) is some measure on JiT. We denote the corresponding r a n d o m stream by r/(A, S). A r a n d o m point set 8' c S generates a substream t](A,S') ^ ?j(A, S). The parametric measure A'(A)- of S' is also absolutely continuous with respect to v(A), with density X' (t) iS X(t) almost everywhere relative to the measure v(A). The passage to S' c: S can be carried out by means of a screening (thinning out).operation, which is defined in the following way. With every point t e T, we associate an event A, or its complement A,, that is, we assume that it is known for each co e Q whether the event A, or A, occurred. We consider (for each co) the subset S(s/) c S consisting of those points seS f o r which the event As occurs, assuming that this screening process yields a r a n d o m point set Since the parametric measures of the r a n d o m streams r\{A, S(s/)) and t](A, S) satisfy the inequality r/(A, ^ rj(A,S), their parameter functions X(t) and X(t,At) relative to the measure v(A) satisfy almost everywhere the inequality (2.1)

X(t,At)^X(t).

In most of the problems which the a u t h o r has investigated, the inequality (2.1) can be extended to all values of t e T. Assuming that (2.1) holds for all t e T, we arrive at the following definition.

5

FIRST PASSAGE

DEFINITION 2.1. The conditional probability of the event A,, the condition being that t e 8, is defined by the relation

(2.2)

P

*{At\teS)

XM^A_

=

If the random streams t]( A, S( t + g, Ak t{v) = {?/[(/, t + «)]} = k; then in accordance with (2.2),

(2-4)

(pk{v) =

is the conditional probability that in a time interval of length v exactly k points from 8 will appear, the condition being that the initial moment of the interval also belongs to S. The / 2 ) v 1 ( d i , ) v 2 ( d i 2 ) .

We assume further that with probability 1 all the points s„ = (sl x, s2yX) e 8 have different first coordinate, that is, ^ a 2 implies =/= Si)Ct2. Considering the random point set S1 = consisting of the coordinates ^ „ of all points sx — {sliX, s 2 J e S, on the space \T1, JtT J , we find that its parametric measure is given by (2.6)

At(At) = £

MMM^i),

A

6

SIXTH B E R K E L E Y

SYMPOSIUM:

BELYAYEV

where ^ ( / j ) = h ) v 2 ( d t 2 ) - Thus, in accordance with (2.2), the conditional probability that for a point (s 1>£t , s2 a) e 8 we will have s 2 x e A2 e the condition being that Sj a = t 1 , is given by

(2.7)

P * ( s 2 , * z A2 1 "

-

'

1

-

J A, 2

X

2

{ t

u

t

2

) \

2

{ d t

2

)

Let us return t o the problem o f determining the distribution o f the derivative, at the moment o f leaving level u, o f the trajectory o f a stationary d i f f e r e n t i a t e Gaussian process = 0, E £ 2 = 1, E £ 2 = k 2 . Here Ty = R\ T 2 = R + = [0, 00), and the J t T i are the algebras o f Borel sets. T h e point „ = x is the instant o f leaving the level u, and s2 a = x is the value £ = x . Here X

f

x

2

w2"^

with respect to Lebesgue measure in T. Consequently, in accordance with (2.7), (2-9)

P * ( 4 > v) =

expj-

that is, the conditional distribution o f the values o f the derivative of a stationary Gaussian process is a Rayleigh distribution ( [ 2 2 ] ) . Let us consider a pair o f correlated random objects (S, £,), where S is a random point set on a space [ T , M j \ having a fundamental system and is a random process (field) with values from a phase space [ 0. Jo To determine a probability measure on the space of trajectories which exit at time t from the point x 6 rr, we put tt(t) = t + th Ai v} = exp rt oo

i

oo

(2.16)

BELYAYEV

0

Xk dF(X),

X2 = 1

l 12 -

3. Random point sets generated by random fields With every trajectory of a random field t e Rm, continuously differentiable with probability one, one can associate various random point sets. The problems which arise here are more diverse than problems concerning the crossing of levels by trajectories of random processes (see Longuet-Higgins [12], [13]). If we proceed along the path of direct generalization of level crossing problems, involving the trajectories of random processes, to the case of random fields, then we arrive at the necessity of studying random point sets Su = {t: £, = u}, which for m = 2 form a family of contours (level lines) in the plane. Problems related to the study of the distribution of the number of such contours, their lengths, and so forth, present difficulties. The approach discussed in Sections 1 and 2 can be used to study local objects, while the level lines formed from the curves = u are not local objects. However, here one can study random point sets which are introduced on the level lines in a special way. A typical example is the study of the random point set S 0 of stationary points of a random field t e Rm, that is, the set of points at which the gradient of the field equals zero. The study of subsets of S0, such as the sets S+ of local maxima (bursts), 0, e > 0,

then with probability one the fields d1{,Jdtidtj have continuous trajectories for which (3.11) is satisfied for any s > 0 and any integer t = km. The complex of conditions k = 1, 2, • • • , holds for We will denote the positive (negative) definiteness of a matrix A by A >- 0 (A -< 0), and also put V£,V' = \\d2(,/dtictj\\. We introduce the following functions:

Theorem 3.4. If the complex of conditions is satisfied, then the kparameter functions with respect to Lebesgue measure in Rm of the random point sets of stationary points S0 = { t } , of bursts S+ = { t } , and of local minima = {t} generated by the random field t e Rm, at which > u, are given respectively by (3.20)

... X°0//1 u(t

>

/* *)

U,gU,>E[l,)t]

E Ì n |detVf,V'| C,i = « „ V k = 0 , t e [ l , f c ] j (.¡=1 pti...p(u1,

(3.21)

• • • ,uk,Q)du1

• • • duk,

X:(t\---,t

-I

UiSu.isIl.k]

E

n i d e t VC(iV'|/+(VCtiV' (.¡=1

=

U,

V£,. = 0, i e [1,

p,i...,k(u1 ,•• - , uk, 0) du± • • • duk, (3.22)

X~(t\

•••, VCt+tiV' £ Ch i e [1, w]. If the conditions k = 1,2, ••• are satisfied, then the parameter function of S'+ is given by

f o r EC,

m

(3.27)

l

t

f

Ai... P .(i4 0 > i4 J ) B | j C„»e[l,»]) =

m

i

|det K J I p o . » ' . - A u >

J Go

U,

Z ) d u d Z ,

where (3.28)

O0 = {mq £ Aq, U, £ A^, (w'i, * • • , Um) G .BJ, ||

du = duQ dui • • • dun,

U = ||u}||,

fa

|| £ Of, i € [ 1 , w],

= ||m]i(?||,

uf = 0.

The special conditional probabilities in the space of trajectories of the field having bursts at the point T = 0, of height he A0, such that e A ¡, and so forth, are introduced as the ratios (3.29)

P l . .

t n

{ A

h B i

, ^

e

[

l

l

W

] M

0

)

=

B

C

i £ \ y , n - \ )

Interesting results were obtained in a paper by Nosko [16] on the structure of excursions of homogeneous Gaussian random fields above unboundedly increasing levels u f oo. It was shown that when the complex of conditions C^ k ,k = 1, 2, • • • , is satisfied, then with probability arbitrarily close to unity an excursion of a trajectory of the field t e Rm, containing a burst at the point T, can be approximated within o ( l / u ) by that part of the second order surface (3.30)

2 =

u

+

\(t

t)'Au(t -

-

m+1

lying on the plane z = u in the space R (3-31)

t)

of points (tlt • • • , tm, z), where ktJ = ® f £ f i -

A„ = « - l l - A ^ o H ,

This result was obtained by introducing in the space of trajectories special conditional probabilities analogous to (3.29). For the particular case m = 2, the excursions are approximated by segments of an elliptical paraboloid. We mention here the following result. THEOREM 3 . 5 (Nosko [ 1 6 ] ) . Let t e R2, satisfying conditional

the

complex

probabilities

of conditions of the following

C^k,

k

=

functionals

be

a homogeneous

1 , 2 , • • • . Then of excursions

Gaussian

field

f o r u ] o o the

special

above

the

level

u:

FIRST PASSAGE

15

the maximum mu of the excursion, its cross sectional area S„ and its volume Vu, satisfy the relations (3.32)

lim P*{umu > v) = lim P*{yuSl U t 00 ut 00

> v}

= lim P*{(2yu3 F J 1 / 2 > v} = exp i - ^ - l , where (3.33)

y = (27tAo)" 1 Ui, 1*2.2 - ¿i,2} 1/2 -

4. Estimation of the distribution of the maximum of a random field Problems concerning the distribution of the maximum of a random field when t e G, where G is a region or surface in Rm, are interesting. An attempt to obtain the exact solution for such problems meets with considerable difficulties. The situation, however, is made easier by the fact that frequently one only needs to find an estimate for the probability of a trajectory of the field going above a high level u. In such cases, one can use an asymptotic approach [1], For example, if C„ t e Rm, is a real Gaussian random field, and S+ is the random point set of bursts of above the level u, then for a region G e Rm with a smooth boundary 8G we have (4.1)

P{sup C, > u] g E r i ( G ) + E>/a+(c)G), leG

where Er/^ (G) is the average number of bursts in S+ n G, and Etj* (dG) is the average number of bursts above the level u of the field t e dG. One can show that if the complex of conditions C ; 1 is satisfied, then the principal term in an asymptotic expression for the right side of (4.1) for u f oo has the form V, where V is the volume of the region G and can be found from (3.25). Thus, the level Uj, for which P{sup (eG C, > ux} « a can be found from the simple transcendental equation

where (4-3)

Cm =

[det A(1>]1/2 2j[im+ 1)12^12-

The results of the numerical computations which were mentioned in Section 3 show that, presumably, the ux obtained from (4.2) gives good results if o > 3. However, the problem of estimating the error which occurs here has not yet been solved. If one assumes that the random stream of bursts above an unboundedly increasing level u is Poisson, then following the method of Cramer [7] one can obtain that for G^R™

16

(4.4)

SIXTH

BERKELEY

SYMPOSIUM:

lim pjsup i < v ^ F v = n«t® VA0 +

+

BELYAYEV

( m

~;)l0gl0 2 V 2 log V

g F

(m - 1) log ( V ^ ) + log Cm + z] 1 = exp { - exp { - ,}}.

As G increases, one has to assume that dG is regular, that is, the volume Vd of the set of points whose distance from G does not exceed d is such that Vd/V -> 0. The relation (4.4) also can be used to calculate the critical level ux. Denoting by Zp the quantile of the level /?, exp { — exp { — z^} } = /?, 0 < /? < 1, we obtain from (4.4) (4.5)

V^u. - V ^ g T

+

( m

-1)l0gl° 2 v 2 log V

g F

(m - 1) log ( v ^ ) + log Cm +

Zl_.

y/2 log V It should be kept in mind that for small a (a S 0.05) and small values of V, calculation of the critical level by means of (4.5) can yield a nonmonotone dependence upon V. The method of calculation by means of (4.2) appears to be preferable. By methods similar to those mentioned in Section 3, we have derived expressions for the parameter functions of the random set of critical points on the level lines of fields defined on manifolds [18] in Rm [1], These expressions can be used to estimate the distribution of the maximum of a field defined over a surface.

REFERENCES

[1] Yu. K. BELYAYEV, "Distribution of the maximum of a random field and its application to problems of reliability," Izv. Akad. Nauk SSSR, Tehn. Kibernet., No. 2 (1970), pp. 77-84. [2] , "New results and generalizations of problems of crossing type," in the Russian edition only of H. Cramer and M. Leadbetter, Stationary Stochastic Processes, Moscow, Izdat. "Mir," 1969, pp. 341-378. [3] , "The random sets of bursts and shines of random fields," Proceedings USSR-Japan Symposium on Probability, Novosibirsk, 1969, pp. 11-18. [4] , "On the characteristics of a random flow," Theor. Probability Appl., Vol. 13 (1968), pp. 543-544. [5] Yu. K. BELYAYEV and V. P. NOSKO, "Characteristics of excursions above a high level for a Gaussian process and its envelope," Theor. Probability Appl., Vol. 14 (1969), pp. 296-309. [6] E. V. BULINSKAYA, "On the mean number of crossings of a level by a stationary Gaussian process," Theor. Probability Appl., Vol. 6 (1961), pp. 435-438. [7] H. CRAMER and M. LEADBETTER, Stationary and Related Stochastic Processes; Sample Function Properties and Their Applications, New York, Wiley, 1967. [8] R. M. DUDLEY, "Gaussian processes on several parameters," Ann. Math. Statist., Vol. 36 (1965), p p . 7 7 1 - 7 8 8 .

[9] A. YA. HINCIN, Papers on Queuing Theory, Moscow, Fizmatgiz, 1963.

FIRST

17

PASSAGE

10] M. KAC and D. SLEPIAN, ' .Large excursions of a Gaussian process," Ann. Math. Vol. 30 (1959), pp. 1215-1228.

Statist.,

1 1 ] D . K Ö N I G , K . M A T T H E S , a n d K . NAWROTZKI, " V e r a l l g e m e i n e r u n g e n d e r E r l a n g s c h e n

und

Engsetschen formein, eine Methode in der Bedienungstheorie," Angewandte Mathematik und Mechanik, Vol. 5 (1967), pp. 1-123. 12] M. S. LONGUET-HIGGINS, "The statistical analysis of a random, moving surface," Philos. Trans.

13] 14] 15] 16] 17] 18] 19]

Roy.

Soc.

London

Ser.

A,

Vol. 249 (1957), p p . 3 2 1 - 3 8 7 .

, "The statistical geometry of random surfaces," Hydraulic Instability, Symposium in Applied Mathematics, Vol. 13 (1962), pp. 105-143. K. MATTHES, "Stationäre zufällige Punktfolgen," Jahresbericht d. Deutsch. "Math. Verein, Vol. 66 (1963/4), pp. 66-79. J. A. MCFADDEN, "On the lengths of intervals in a stationary point process," J. Roy. Statist. Soc. Ser. B, Vol. 24 (1962), pp. 364-382. V. P. NOSKO, "The characteristics of excursions of Gaussian homogeneous fields above a high level," Proceedings USSR-Japan Symposium on Probability, Novosibirsk, 1969. C. PALM, "Intensitätsschwankungen in Fernsprechverkehr," Ericsson Technics, Vol. 44 (1943), pp. 1-189. P. K. RASEVSKII, Riemannian Geometry and Tensor Analysis, Moscow, Gostehizdat, 1953. S. O. RICE, "The mathematical analysis of random noise," Bell System Tech. J., Vol. 23 (1944), p p . 2 8 2 - 3 3 2 ; Vol. 24 (1945), p p . 4 6 - 1 5 6 .

20] C. RYLL-NARDZEWSKI, " R e m a r k s on processes of calls," Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Berkeley and Los Angeles, University of California Press, 1961, Vol. 2, pp. 66-79. 21] E. SLUTSKY, "Qualche proposizioni relativa alia teoria delle funzioni aleatorie," Oiorn. 1st. Ital. Attuari, Vol. 8 (1937), pp. 183-199. 22] V. I. TIHONOV, Excursions of Random Processes, Moscow, Izdat. " N a u k a , " 1970. 23] F. ZITEK, "On the theory of ordinary streams," Czechoslovak Math. J., Vol. 8 (1958), pp. 448-459. (In Russian, summary in German.)

LIMIT THEOREMS FOR RANDOM WALKS WITH BOUNDARIES A. A. BOROVKOV INSTITUTE OF

MATHEMATICS

NOVOSIBIRSK,

U.S.S.R.

1. Introduction In this review, we consider boundary problems for random walks generated by sums of independent items and some of their generalizations. Let £2, • • " be identically distributed independent random variables with distribution frunction F(x). Let S0 = 0, Sn = ~£nk=£k with n = 1, 2, • • • . We shall study the properties of the random trajectory formed by the sums S0, Slt S2, ' ' ' • Let n be an integer parameter and let g*(t) be two functions on the real axis with the following properties: ( 1 - 1 )

0

>

flf-(O),

5C(t)

> gn(t),

t ^

0 .

We shall denote by On the part of the halfplane (t S; 0, x) which lies between these two curves. In the same halfplane (t, x), let us consider the trajectory formed by the points (1.2)

(s'5*)'

fc

= 0, 1,2, •••

One of the main boundary functionals of trajectory (1.2) is the time rj c at which it leaves the region Gn: (1.3) We shall define the value of the first jump yG across the boundary of the region Cr„ by the equalities (L4)

XG = Sno - fif„+ (f/ c )

or

XG =

-

g~(riG),

depending on whether trajectory (1.2) crosses the upper or lower boundary of the region G„. Note that in general the random variables r\G and %G are not defined on the whole space of elementary events. We put r]G = 00, where rjG remains undefined. We shall not define the random value XG on the set {t]G = 00 }. Problems variously connected with distributions of the functionals rjG and XG will be called boundary problems for random walks. It is well known that these problems play an important part in mathematical statistics (in sequential analysis, nonparametric methods, and so forth) in queueing theory, and in other 19

20

SIXTH B E R K E L E Y SYMPOSIUM: BOROVKOV

similar fields of mathematics. For instance, the classical problem of sequential analysis leads to the elucidation of probabilities of the type (1-5)

P(XG

> 0;r,G


x)

2m+1c+e = , lm "

+

* (a; + i|a|w)"

where n0 = n0(m, cm) is known explicitly. Concerning the asymptotic properties of the distribution of S, it is well known that if X + > 0, (p(X+) ^ 1, then P(S > x) ~ c1e~qx with x -* oo (at X+ = q we also suppose that —co,thenP(S > x) ~ c2H(x), where c2 = for X+ = 0, and where H(x) = J™ (l — P{t)) dt. It is assumed here that for any 0 < h ^ 1 and t oo, (4.6) v '

exp n

{W +

} m „ + h ) - 1, H(t)

0
0 such that a(t) = 1 when t > 0, a(t) dt < co. By itself, weak convergence, F1 => F2, is not sufficient for the convergence of suprema distributions. The class 0t, mentioned above, contains all the distributions for which either 0, t e [0, 1]. If inf ( 0 1 ) g(t) < 0, then P(t]G ^ 1) -» 1, when n —> oo (here again E£k = 0) and we shall investigate the rate of convergence to zero of the complementaiy probability P(t]G > 1). The asymptotic nature of this probability appears to be quite different and significantly more complicated. Here one can find only the asymptotic behavior for the logarithm of this probability P{f]a > 1 )•

RANDOM W A L K S W I T H

BOUNDARIES

27

Suppose g(t) has no discontinuities of the second kind and for every t, g(t) = min (g(t — 0), g(t + 0)). Let t0 be the point where i n f ( 0 1 ) g ( / ) = gf(i0) and let B0 be the set of points (0, 0), (t, y > g(t)) for 0 ^ t ^ t0. We denote by h(t) the lower boundary of the convex closure of the set B0. The curve h(i) evidently realizes the shortest path from point (0, 0) to point (t0, g(t0)) which does not intersect B0. This function will be convex and absolutely continuous. If we put — 9(h) o n [ co. (As already noted, one can reduce the case x ~ en to x ~ n by changing the boundary.) THEOREM 5 . 1 . If h'(0) > — oo and the interval (h'(0), 0 ) contains no point of discontinuity of the function A(a), then log P(rjG > 1) ~ —nW(h). To describe the asymptotic form of the distribution r\G in the problem with a fixed terminal value, that is, the probability of the event {r]G > 1, SJxe Aj,} (where Afc is the neighborhood of the point b), it is necessary to construct the shortest path hb(t) connecting the points (0, 0) with (1, b) and not intersecting the set of the points (/, y > g(t)), 0 ^ t ^ 1. (This path will evidently coincide with h(t) in the segment [0, < 0 ] if bt0 ^ g(t0).) THEOREM 5 . 2 . If Ab is a shrinking neighborhood of the point b, (5.2)

lim lim ^ log P( r,G > 1, ^ e A t ) = - W(hb). Ab-0 n->oo n y X J

If is lattice-like and P(S„ = bx) > 0, then from the very beginning we may take a single point b as a A;,. Now let us consider the case x = o(n). Introduce the functional (5.3) THEOREM

(5.4)

V(h) = | W 5.3.

If x/n

2

dt =

0, x(n log n)

1/2

dh „ — dh(t). n at 1

-> oo, with |A±| > 0, then

logP(f/ G > 1) ~

The analogous result is true for P(r\G > 1, SJx e Ab). These theorems, obtained in [10], show what the exponential part in the probability P(rjG > 1) is. As for the power factor, we may conclude on the basis of different examples that it may be of any form depending on the corresponding function g(t). It is possible to calculate this factor only for very special kinds of boundaries g(t), for example, for the broken lines consisting of a finite number of straight line segments.

28

SIXTH B E R K E L E Y

SYMPOSIUM:

BOROVKOV

6. Generalization It is natural to consider the next generalization of these problems. For example, some asymptotic problems of testing theory for statistical hypotheses lead to this. Denote by Sn(t) a random broken line connecting the points (k/n, Sk/x) for k = 0, 1, • • • , n. In the previous parts, the asymptotics of (6.1)

logP^iOeG)

were investigated when G is the set of functions/(í) < g(t), t e [0, 1], The question concerns the behavior of (6.1) when n -* co if G is the arbitrary open set in the space C(0, 1). The following theorems hold here [10]. Denote by G the closure of G in the metric of space C(0, 1) and by E a 0(0, 1) the set of all absolute continuous functions /(£), /(0) = 0, for which there exists a finite number of intervals A 1; A 2 , • • • , Ar; u Aj = [0, 1], where/'(¿) is monotone and bounded. Obviously, A(/'(G feEnG

inf

feEnG

inf

feEnG

P (f'(t))2 J0

Let x ~ n and W(G) = W(G). If log P( 0, THEOREM 3 . 1 .

(3.1)

P[|i?„ - ERn| ^ eER„] =

0((p(n)n-2rn'2).

Let a > 1 and {%} be as above. Then by the Borel-Cantelli lemma, RJERn —• 1 a.s. along this subsequence. To fill in between, by the monotonicity of Rn, for nk ^ n < nk+1,

R A N G E O F RANDOM W A L K

,3 2)

"« ERnX < < + ERnkERnk„ -ER„-ERnk^ R

35

n , ER„k '

ER k +

and we know that the lower and upper bounds converge to a " 1 and a, respectively, since ER„ ~ nr„ and rn is slowly varying. But a can be chosen as close to 1 as we please, so this is sufficient. The remainder of this section will be devoted to obtaining the desired estimate for Var R„.

For any recurrent random walk in two dimensions, Var Rn =

THEOREM 3 . 2 .

0(cp(Sn)) where

¿-'(0, x)Px[Tx >i,T0> xtO

i].

Hence, we can write (3.7)

Cov (Z„

Zj) = EZtZj - r,rj = £ ^¿"'(0, x)b,(x), x?0

where (3.8)

bt(x) = PX\TX >I,T0>Q-

PX[TX > QPX[T0 > I]

= PX[TX ^ i, T 0 g q - Px[Tx S qpX[T0

g q.

Using the last expression for b^x), we will find a more useful form for it. We have

(3.9)

PX[TX i - v -

T0 > i - v - {y> 0) xfOy+0 x+0

+ M2}

p^xyiyEixtf

S2 yp-2E\XA2

= O

1 n log n

The proof is now complete for 1

(4.8)

x+0y+0

^B

C1P

n log n

the last step being a consequence of the known limit r„ ~ ct/log n. (See Lemma 2 . 3 of [7].) THEOREM 4.2. For two dimensional random walk with EXx = 0 and E\X,|2 < oo,

where c2 = 87t2Ji|Q|, the covariance (4.10)

c->n2

Var R„

(4.9)

K

=

log w -

o 1 - w +

w2

log n

matrix of dw

+

is Q, and

I n 2

12

= 0 84948659

Note, however, that if the random walk takes place on a proper subgroup of E2, then a transformation should be made so that it will take place on all of E2 and Q should be the covariance matrix for this transformed random walk.

42

SIXTH B E R K E L E Y S Y M P O S I U M : J A I N A N D PRUITT

PROOF. The first step is to make a transformation, if necessary, so that the r a n d o m walk does not take place on a proper subgroup. This does not change R„ and the fact that Q may change has been allowed f o r in the statement of the theorem. W e shall assume f o r now that the r a n d o m walk is strongly aperiodic and show at the end how to remove this assumption. F r o m (3.5) a n d (3.7), we have that n n—/ (4.11) Vari2„ = 2 £ £ £ ¿»¿(0, x)bt(x) + 0(n). ¡=1 A =1

The expression we will use f o r bt(x) is given in (3.18). The contribution t o the variance f r o m the negative part of b ^ x ) will be considered first. This part will be denoted — V„ and the positive part U„ so that Var R„ = Un — V„ + 0(n). Now (4.12)

F„ = 2 t

Z

¡=1

¿=1

= 2 £

"z

E E x+0 S=1

p

t

t i

o(0, x)P&x,

W v

¡=1 A = 1 I=L V=I- n/log 5 n, one can replace log £ with log n. The same method applies to the other log terms in the denominator by considering where i, i — and n — i + £ are respectively n/log5 n. Thus, (4.13)

V„ ~ 2c?(log n ) - 4 £ i =

log " ~ ' + * log 2 4=2 S »—€

n

z

2

2

= 2c 1n (\ogn)-*^(j

)

1 _ log -

l),

Z

y

+ y

z

log

z —y

dy dz

since the double integral can be evaluated. N o w we must consider the positive contribution. In (3.18), write (4.14)

PX\TX > i - v - I T0 > i - v -

{]

= r , _ v _ { - PX{TX > i - v -

T0 ^ i - v -

£];

the next step will be to show that using the last of these probabilities in (3.18) will lead to a term of smaller order. T o obtain a b o u n d , note that by thinking of t] as the last time zero is hit prior to i — v — i - v - 4

(4.15)

PX\TX > i - v -

T0 ^ i - v - Q ^

X ¿"A*, O ^ - v - « - , , . 1=1

R A N G E OF R A N D O M W A L K

43

Substituting this bound in (3.18) and (4.11), we see that this contribution to the variance is of order (4-16)

t 1 I Y,llPo(0,x)Pl(x,0)Pl(0,x)P^x,0)ri^.^n ¡=1 A=1 x? 0 S + v + r/gi ^ l Y I I,Y,T,Pi(0,x)Pt{x, 1=1 A = 1 xf 0 i + v + lf^l

0)i»(0,

x)P"(x,0)rn_i_xrtv_^

= o(nr5„t S S E E ^(O^lP^x.O^lO^lPV.O)), \ A=1 x + 0 i + v + ijSn-A / where Lemma 2.2 has been applied four times at the second step and the i sum has been moved inside and Lemma 2.1 used at the last step. We will now show that the multiple sum is of order n. Since it is symmetric in A and v, in { and t], a bound is 4A2

(4.17)

f

f

X

t

Px(0,x)PHx,0)v~1r1

Z

A=1 £=1 i f O v = A 11 = 4 . . . 2 V V A =1 i=l "v

1

1

etl

m

1

A

C en

,

Thus, we have for the positive contribution to the variance (4.18)

Un = 2 £ Y I t "l i=l A = 1 x^O i=l v — 1

0 ) ^ ( 0 , x)r ( _,_ 4 +

0(n2r5n).

We will now apply Lemma 2.2 three times to the main term here to obtain n n—i i i—£ (4.19) 2 £ I E £ X P*(0,x)PHx,0)Pv(0,x)rn. i=l A= 1 1^0 5=1 v= 1 But since this is now the leading term, we need to examine the error made in the application of the lemma. To this end, note that for y ^ 1, m—k m—k (4.20) 0 ^ X fj(ri-k-j - rl_k) < y £ i_ J -(r m _ k _ J . rm.k). j=i J= i Now, for 1 g j (4.21)

rm_k_j

(m — k)/2, by Theorem 4.1, - rm_k =

m-k , £ ft = O = m — k — j+ 1

j \

m

r m-k b ~~ k

44 and so (4.22)

SIXTH BERKELEY SYMPOSIUM: JAIN AND PRUITT m—k

X SMm-k-! j= i

-

rym-u)

(

(m-k)/2

m-k

j=Z 1

ifj(m-k)-1r£}k+

= 0(r>m+-l + r>+_\) =

}=

£[m-k)/2 f j , k - k - j

0(rV-\).

Utilizing this in (2.12) and (2.11) yields (4.23)

m

X P k o(0,xy m _ k =

k= 1

fc

m

X P k { 0 , x ) [ r l + _ \ + 0(rJ, + 4)]. =1

Thus, the error introduced by Lemma 2.2 will always lead to an extra factor of an r and this will ultimately give a term of order w 2 /log 5 n. We now return to the estimation o f the main term (4.19). By summing first on i, we obtain (4.24)

C7„ = 2 X akbn_k

+

0

/

^ log 5

where (4.25)

ak=

X Z Z i + v+A = k x+0

(4.26)

6k =

Z rjri.j ~ j=o

°)pV(°' fcrj.

We will prove next that ak -> a; this will imply that (4.27)

Un~2a

£ bn_k ~ an2r4n ~ ^ J - . *=3 log 4 M T o show that ak -> a, the local limit theorem ([12], pp. 77 and 79) will again be used. Write (4.28) uniformly in x as

P'(0, x) = Qfi(x) + ft

o(Ep(x)),

oo, where

(4.29)

Qf{x)

= (2nP)-1\Q\~i'2

(4.30)

Ep(x)

= min {j9 _ 1 , | x | - 2 } .

Since it is clear that Qp{x) = 0(Ep(x)), to zero if (4.31)

exp

the sum of all the error terms will go

Z Z Z Ex(x)E4(X)Ev(X) i + v + A = k xi=0

is bounded. Since this is symmetric in v, A, we may as well assume that £ ^ v ^ A. This means that A k/3, and so (4.31) is bounded by a constant times

45

RANGE OF RANDOM W A L K

(4.32)

r ' i i

X

i = l v=l |x|2Sk

^

k

~1

Et(x)Ev{x)

l Z |x|2glk \

+ k2

£

|x|2gk

H"6

e 2 k^ 2 g ^ ) + PI

o

[

, 2 log

fcl/2

j=i

ek1/2\2 J

= 0(1).

Now we must examine the leading terms. Letting (4.33)

n = r

+ v " 1 + (k - £ - v ) " \

1

y =

xfi1'2,

we can write (4.34)

ak =

£ £

X

+

,5 + v + /l = Jt x^O

= (27t)- 2 |g|- 1 V

i=l

c

-2

Z

o(l)

Z {£v(& - £ - v ) } - 1 © , ^ ) + o(i)

1

v=l

y

k-2 /i-i- 1 X z {iv(ft - 5 - v)//}" 1 ~ i = i V=1

20:%,

where (see [5], p. 533) (4.35)

1 r K^ = -

ri-z

2 Jo Jo

j

i

z - z + y -y 2

, , dydz = -

2 -yz

2

r

iogw 2

dw

)0 1 - w + w

2

= 1.17195361935 Recalling (4.13) and (4.27), (4.36)

Var Rn ~ 2cf ^

+

-

n2(log »)"*.

Thus, the proof of the theorem is complete for the case when the random walk is strongly aperiodic. To make the transition to the general case, let Sn denote the original random walk which has been made aperiodic by a transformation and let P{x, y) denote its transition function. Then the random walk S'n with transition function (4.37)

P'(x, y) =

y) + \P{x, y)

will be strongly aperiodic. One can describe the paths of the new random walk by flipping a coin and each time remaining stationary if the coin falls heads and moving according to the original transition function if it falls tails. The range of the S'n random walk will be denoted by R'„. By conditioning on the sequence of heads and tails, it is clear that (4.38)

ER'n =

ZQ Q

2~"ER k ,

E(R'n)2 =

Q

2~"ER 2 .

46

SIXTH B E R K E L E Y SYMPOSIUM: J A I N A N D PRUITT

For even n, (4.39)

| E R 2 2 - ER2k\ = |

£ £

EZiZj

-

£ £

tfZ^I

< w|w — 2/cI Thus, by (4.38), (4.40)

|S(I?;)2 -

2 ^ 1

^

2~"n\n - 2k\

X

|n — 2fe| i n ! "

+ < n 5 / 3 + w2P

2~"\ERl

Y -

- >

2 ~ 2

- ER2nl2\

-n2'3

where Y is a binomial random variable. By Chebyshev's inequality this probability is 0(«~ 1 / 3 ). Thus, we have (4-41)

E(R'n)2 = ER2n/2 +

0(n5'3).

The same estimate can be obtained for (ER'„)2 and (ERn/1)2 same way. Therefore, (4.42)

Var R„ ~ Var R'2n

in essentially the

Sn2K\Q'\4:n2 log 4 n

by the strongly aperiodic case. But since Q' = Q/2, it follows that \Q'\ = |©|/4 and the constant has the right form. 5. Sufficient conditions for strong transience The following theorems give simple sufficient conditions for a random walk to be strongly transient. As we already mentioned in the introduction, the results of [6] apply to the range of such random walks. First we need to introduce some terminology. A nonempty set F c Ed is said to be closed if hyeF P(x, y) = 1 for every x s F. The random walk is called irreducible if the only closed set is Ed. This is equivalent to saying that every lattice point can be reached from every other lattice point. The assumption that the random walk is aperiodic (that is, does not take place on a proper subgroup oiEd) which we have made without any real loss of generality is equivalent in the present terminology to the assumption that no two closed sets are disjoint. THEOREM 5 . 1 . Every aperiodic random walk that is not irreducible is strongly transient. PROOF. Let = {x: 0(0, x) > 0 } and F0 = {x: G(x, 0) = 0 } . Since the random walk is not irreducible, F1 is a proper subset of Ed and hence F0 is nonempty. The sets Fl and F0 are both closed and by the aperiodicity must have a common element x. Then there is a A: such that

R A N G E OF R A N D O M

P k (0, # ) = £ > ( ) ,

(5.1)

47

WALK

G(x, 0) = 0.

We may also assume that £ < l , f o r i f e = 1, then un = 0 for n ^ k by (5.1) and the random walk is clearly strongly transient. Since Pk(0, y) ^ 1 — £ for y =j=- x, Pk(0, y) ^ a = max (e, 1 - e)

(5.2)

for all

yeEd.

Now suppose that zeF1. Since G(x + z, z) = G(x, 0) = 0 and G(0, z) > 0, it follows that G(aj + z, 0) = 0 or x + z e F0. Thus, for zeFu £ Pk(z, y) g 1 - Pk(z, x + z) = y±Fa

(5.3)

e^oi.

l -

Now suppose for all Z 6 I , n Fc0 that Pnk(z, 0) ^ a"; this is true for n = 1 by (5.2). For zeF,. nF°0, then (5.4)

p(n+1)k(z,

o) =

x p k ( z ' y)pnk{y> °) ^ yeFinFo

a

" Z pk(z> y) ^ HFo

a +1

"

by the induction hypothesis and (5.3). If 0 £ F0, the random walk is trivially strongly transient so we may assume that 0 e Ft n Fc0. Thus, we have u„k ^ a" and for 0 < r < k (5-5)

«„*+,=

Z Pr{0,y)Pnk{y, jieFinFo

0) ^ a";

this is clearly sufficient for strong transience. 2 THEOREM 5.2. A random walk with EXi =/= 0 and E^^ < oo is strongly transient regardless of the dimension. PROOF. Since EX1 ^ 0, there must be at least one component of X1 with nonzero expectation. Furthermore, whenever the original random walk visits 0, so will the component ones so that w„ ^ u'„ where u'„ refers to one of the component random walks. Thus, if the component random walk is strongly transient, the original random walk will be as well. This reduces the problem to the one dimensional case. Let S„ be a random walk with EX1 0, EXj < oo, and d = 1. If Var Xj = 0, the random walk is degenerate and EX^ ^ 0 implies u„ = 0 for all n > 0. We may then assume that 0 < Var Z j < oo and by considering — Sn, if necessary, that fi = EXx > 0. Now (5.6)

X

f>

n= 0 k= 0

n +t

=

£

E E H 0,X)P^,0) =

n = 0 k=0

x

^ ( 0 , * 0 ) x

= G2(0, 0) + 2 X

G

(°> x)G(x, 0).

x> 0

Thus, the random walk is strongly transient if and only if the last sum converges. If the random walk is aperiodic, the renewal theorem asserts that l i m ^ ^ G{0, x) = > 0. If it is not aperiodic, the limit will still be positive if we restrict x to. the subgroup. In either case, the random walk is strongly transient if and only if S x > 0 G(x, 0) converges. But

48

SIXTH B E R K E L E Y SYMPOSIUM: J A I N A N D PRTTITT

X

(5.7)

G X

x> 0

( >

0)

=

I

«(0'

*) =

E

x< 0

= £

Pism

n=0

F

E

"(0> X)

x 0 let A(t, s) be the solution of (1-6)

f ( x , t) = e.

Then (1.7)

P{w(t)

^ A(t, e) for some t ^ t} = P{f(w(t),

t) ^ e for some t ^ r},

and we use an elementary martingale equality to evaluate the right side of (1.7). The relation of w(t) to the sequence of sums of i.i.d. random variables with mean 0 and variance 1 then permits the asymptotic evaluation of boundary crossing probabilities for partial sums. In view of (1.3) a choice of F of particular interest is, for S > 0, -1 (1.8)

lOft-! J )

dF(0) = •

(loft^

dQ

for 0 ^ - ,

0

otherwise,

for which it is shown in [ 1 1 ] that for any e > \/5, (1.9)

A(t,s)

=

21[ log 2 t + f l o g j t + X log it ¡=4 1/2

+ (1 + ''' i-i-d. random variables. W e begin in Section 2 by establishing an analogue of the criterion (1.2) for a law of the iterated logarithm for sample minima. In Section 3, we give an application of this result to a conjecture of Darling and Erdös [ 2 ] , In Sections 4 and 5, we introduce a continuous time process « , , 0 < t < oo, related to min (x^, • • • , x„) in much the same way that w(t) is related to Xj + • • • + xn, and apply the methods of [11] to the study of this process. In spite of the dissimilarity in the behavior of vt and w(t), the measure F defined by (1.8) plays the same role for v, as for w(t).

ITERATED

53

LOGARITHM

2. The law of the iterated logarithm for minima of uniform variables THEOREM 1. Let w 1 ; u2, ••• be independent and uniform on ( 0 , 1 ) , and let V„ = min (Mj, • • • , «„). Let (c„) be any sequence of positive numbers. Then: (i) if cjn I for all sufficiently large n, then P{n Vn ^ c„ i.o.} = 0 or 1 according as °° r

(2.1)

X -

i n

converges or diverges; (ii) if cjn [and c„ 1 for all sufficiently according as 00

(2.2) I

converges or diverges. COROLLARY 1. Fork (2.3)

large n, then P{nV„ Si c„ i.o.} = 0 or 1

c n

3,

p|W7„ ^ log 2 n + 2 log 3 n + £

log; » + (1 +

l o g t + 1 n i.o.

is equal to 0 or 1 according as 5 > 0 or S ^ 0. REMARK 2 . 1 . The proof of (i) is an immediate consequence of the BorelCantelli lemma and the fact that if cjn is ultimately decreasing, then Vn ^ cjn i.o. if and only if u„ ^ c„/n i.o. The proof of (ii) is much harder and will be given below. REMARK 2 . 2 . If Mn = max (ult • • • , un), then Theorem 1 holds with V„ replaced by 1 — M„. REMARK 2.3. Under different regularity conditions on the sequence ( c j , Ville [13] has shown that if (2.2) converges, then P{nVn ^ c„ i.o.} = 0. His approach is similar to the one we take in Section 4. Pickands [8] has also obtained some results in the direction of Theorem 1. R E M A R K 2.4. The condition in (ii) that c„ be ultimately increasing is bothersome in some applications (see R e m a r k 2.5 below), but it cannot be dropped completely. F o r example, if c„ = 1 /», then both (2.1) and (2.2) converge. Hence by (i), P{nVn ^ c„ f o r all sufficiently large n) = 1, which is incompatible with the conclusion of (ii) applied t o the same sequence c„. REMARK 2 . 5 . Let xtl x2, • • • be independent r a n d o m variables with a common continuous distribution function F. Since u„ = F(x„) is uniform on (0, 1) and (2.4)

^ m i n (x,, • • • , *„)] = min [ J ? ^ ) , • • • , F{xn)\ = min ( « ! , • • • , « „ ) = V„,

Theorem 1 implies a law of the iterated logarithm for min (xlt • • • , xn). In particular, (ii) implies that if (o„) is any sequence of numbers such that an is ultimately decreasing and nF(an) is ultimately increasing, then P { m i n (xi, • • • ,

54 x

n

SIXTH BERKELEY SYMPOSIUM: ROBBINS A N D

SIEGMUND

^ a„ i.o.} = 0 or 1 according as

(2.5)

£-F(a„) exp i

{ - n F ( a

n

) }

< oo or = oo.

The condition that n F ( a ) be ultimately increasing may be difficult to verify for a given F, and hence, it is worth observing (as will become apparent in the proof below) that the condition in (ii) that c„( = nF(a )) be ultimately increasing may be replaced by the growth condition n

n

(2.6)

liminf— n->oo log 2 n

Moreover, it follows a f o r t i o r i that if c„ ( ^ ) c'„ and P { n V „ ^ c'„ i.o.} = 1(0), then P{ nVn ^ cn i.o.} — 1(0)- Hence, (ii) may be applied indirectly to some sequences (c„) which satisfy neither the monotonicity conditions of (ii) nor the growth condition (2.6). REMARK 2.6. Let xlt x2, • • • be independent N(0, 1) random variables with distribution function O(aj) = {100 0r) = ^ p f x ) a;

1 +

0 [ \ Var

as

x

-*

—

so,

it can be shown that c„ = wO(a„) satisfies (2.6), and hence, by (ii) and the preceding remark, that P{min (xlt • • • , x„) ^ a„ i.o.} = 0 or 1 according as 5 > 0 or S ^ 0. Alternatively, it is possible using (2.8) to replace the criterion (2.5) by one involving the normal density (p ; the argument of Lemma 8 below (together with (2.5), (2.8) and Remark 2.2) shows that if (a„) is any ultimately increasing sequence of positive numbers such that n a ~ ( p ( a ) is ultimately increasing, then P{max (xx, • • • , xn) ^ an i.o.} = 0 or 1 according as 1

£ < p K )

(2.9)

£

1

a

n

exp

j 1 i

n

n

0 and n = exp {aA:/log k ] , k = 2, 3, • • • , and assume k

that

c j n

is

ultimately

decreasing.

ITERATED LOGARITHM

55

(i) If (2-10)

Zexp{-c[nk]} k

converges for some a, then P{n Vn c„ i.o.} = 0. (ii) If (2.6) holds and (2.10) diverges for some a, then P{nVn c„ i.o.} = 1. As usual, [ x ] denotes the largest integer g x. To avoid burdensome detail in the proof, we have ignored the difference between nk and [n t ]. PROOF. For (i), suppose that (2.10) converges for some a. By replacing c„ by min (c„, 2 log 2 n), we may assume, without loss of generality, that (2.11)

c „ ^ 2 1og 2 n.

By the Borel-Cantelli lemma, it suffices to show that (2.12)

^ cn for some % < n

k

nk+1}

< co,

and hence, by the monotonicity of V„ and the ultimate monotonicity of cjn, to show that (2.13)

k

n

(.

k+1

>
KA (2.31)

c„r ^ A log r.

With K2 = max (Kl,K3,KA), LEMMA 4 .

(2.32) PROOF.

the lemma follows from (2.29), (2.30), and (2.31)

For each k ^ K2 and r >

rlt

^ exp {— A 3 (log r) 1 / 2 }.

^

From (2.29) and (2.31), we have

(2.33)

£

£ exp{-(l -

e x p { - A ^ } ) c „

^ exp { - ( 1 - exp { - A (log r ) _ 1 / 2 } ) A log r) ^ exp {— A 3 (log r) 1 / 2 }. LEMMA 5. For each k, let r2 = r2 (k) be the least integer r > k such that r ^ k + (log r)2. Then for all k > K1 and r ^ r2, nk/nr ^ 1/r. PROOF.

B y (2.27), for k ^ Kt

nt . /r — k\ log — ^ -Aoe ^ - A a log r < - l o g r nr \ log r )

(2.34) LEMMA 6.

There exists a number K 5 such that for all k S; Ks

(2.35) PROOF.

and r ^ r2,

^

and r > r2>

^ Y2P(ArY

From (2.23), we have

(2.36)

P(Ar) ^ A e x p { - c n r }

for all r ^ some K6. Hence, by Lemma 5, (2.11), (2.29), and (2.36), we have, for all k ^ K5 ^ max (Kx, K6) and r ^ r2, (2.37)

P\v n n * ^

^ exp j - h 1 „ .

- ^

|c.

("2 l o g , wJ

1

Note that r 2 (fc) ~ k as k —> oo. Let Kn be so large that for all k Si K7, (2.38)

r 2 (fc) ^ 2k

and (2.39) LEMMA 7.

8(log A;)2 exp {— A 3 (log k)112}


0, let h = h(t) = inf {t : t > z, v, < v t }. Then {h > t} = {vt = v }, and hence, by (4.3), (4.4)

P{h > t\vz = a) = exp {-a(t

- r)}.

Also for a2 < al, (4.5)

P{vh

a2, t < h, < t + 5\vz = = P{vt

+3

at}

^ a2, t < h < t + 5|«t = tti} + o(5)

= P{v,+s ^ a2, vt ^ ai\vx

= at}

+ o(5)

= (1 — exp { — ¿a 2 }) exp { - ( < - t ) a j } + o(5), so (4.6)

P{vh ^ a2\vz = a^} =

r°° JT

P{vh -SL a2, h e dt\vz = a^

a = A

I

It follows that the sample paths of the process vt, 0 < t < oo, may be described as follows: for any T > 0, if the process is in the state a at time T, it remains there for a random length of time having a negative exponential distribution with parameter a and then jumps to a point uniformly distributed on (0, a). By (4.2), P{v0+ = + oo} = 1, and with probability 1 there are infinitely

64

S I X T H B E R K E L E Y S Y M P O S I U M : B O B B I N S AND SIEGMUND

many jumps in each neighborhood of t = 0. Except for the behavior of v, near t = 0, this description is analogous to that of the discrete time process Vn, which holds in each state a a random length of time which is geometrically distributed with parameter a and then moves to a point uniformly distributed on (0, a). If xlt x2, •• • are i.i.d. with P{x{ x} = e~x, and v* = min • • • ,xn), then the process v, interpolates the process v* in the sense that the two sequences vn and v*,n = 1, 2, • • • , have the same joint distributions. Trivial modifications of the proof of Theorem l(ii) prove: THEOREM 3. If c(t) 2 : 0 is ultimately increasing and c(t)/t it ultimately decreasing, then P [v, ^ c(t)/t, i.o. t f oo} = 0 or 1, according as J f (c( 0, and hence, Jo dF(z/t) -> 0 as t -> oo, contradicting o u r supposition. Since F' is decreasing in a neighborhood of the origin, we have by (5.4) for all t sufficiently large, (5.7)

e=f{A,t)

=

and hence, by (1.8) and (5.5),

e»'F'(y)dy

^

F'(A)(e'A t

- 1)

67

ITERATED LOGARITHM „tA

lim sup

(5.8)

< e. t A

l 0 g

A Vl0g2

A

Rewriting (5.8) as (5.9)

A


0 and n ^ xm by n 1.) The proof is similar in spirit to the proof of Theorem 2 of [11], but the details are much simpler.

69

ITERATED LOGARITHM

R E M A R K 6.3. Using the probability integral transform, one can immediately obtain analogous results for random variables having arbitrary continuous distributions. (See Remark 2.5.) F o r example, if x x , x 2 , • • • are i.i.d. with P { X i ^ x } = e ~ * ( x > 0) and if g = e ~ h satisfies the conditions of Remark 6.2 above, then the left side of (6.2) equals

(6.3)

lim P< max x k ^ log ra + h i m— ) for some n

m-* oo

[lgfcSn

\

J

z m

If the x are standard normal random variables, the left side of (6.2) equals (6.4)

lim P ^ max

'1„ = t 0 i + • • • + xn n and since the terms are independent as a result of the process being Markovian, using elementary properties of the Laplace transform, we obtain 1 fa+ ¿00 A„-!(z)e z dz, 4>„(0 = Ini zA„(z)

(2.4)

a > 0.

F r o m formulas (2.2), (2.3), and (2.4), it is not hard to determine the expectation o f T 0 > n a n d T„_ l ! n :

(2.5)

MTO,„ = T0,H = AB>1 = V y ^ I 6S k = 0 AkVk s= 0 1^n-lan-lfl

dk k£= 0 '

T o find the distribution of t„, we use formula (1.6) : n- 1 E Pk k=0 [1 - (!>„(*)] dx, (2.6) 1 - 6n(t) = P{%„ >t} = T„ - i , n

where the pk are the stationary probabilities of the states. Hence, again using the properties of the Laplace transform, we obtain (2.7)

0n(t)

= 1 - 9n +

+

= 1

'a + i ooA„(z) 2 ni o — ioo a + ico

27ti

a— loo

2

A„_!(z)

A„(2)7,„_l

e z( dz

n

0,

where K = (2.8)

c , >

n— 1 I Pk, k=0 A„(z) -

A„_!(Z)

!

2 /

- n- l,n

3. Properties of the polynomials A„(z) and 5„{z) L e m m a 3.1. The roots of the polynomials A„(z) and 0V = P-o [T0,n

e~'.

82

S I X T H B E R K E L E Y S Y M P O S I U M : SOLO VIE V

It is not hard to show that for x — 1

REMARK 5 . 1 .

(5.8)

^„e--i

Gn(T0.,) - 1 +

. . I *

from which it follows that for -* 0 the bound (5.1) is exact, as concerns order. REMARK 5.2. Proceeding as in Theorem 4.2, it is not difficult to obtain a more precise bound. For j8 < \ (5.9)

Gn(t) = 1 -

£ exp { - A j

+

0

_JL_,exp

j_

|0|
„

=

L

7. A class of infinitely divisible laws We have earlier obtained conditions under which the distributions of the suitably normalized variables T 0 J „, T„ and i„_ 1>n converge to an exponential distribution (possibly with a jump at zero). It would be natural to go further and attempt to find other limiting distributions for these variables. If, passing to the limit, one varies the level n as well as the parameters Xk and fik, then the problem is empty, since by choosing these parameters appropriately we can always make the roots of the polynomials An(z) equal to any preassigned negative numbers. Therefore, we assume that the parameters Xk and fik are fixed, and that the level n —• oo. For this statement of the problem, the following holds. THEOREM 7 . 1 . Suppose that the parameters Xk and fik satisfy the conditions (7.1)

lim T0 fi-* 00

n

= +co,

84

SIXTH BERKELEY SYMPOSIUM: SOLOVIEV l i m T^ll

(7.2)

=

,n

lim 2 «-•co X„T. «-* n,n + 1

(7.3)

0 ^ 7 ^ oo.

y,

Then a + i oo lim

(7.4)

t _i = exists. Since T0 n j oo, to find the limit of the a„ k, we can apply a theorem of Stolz [3] (the finite-difference analog of L'Hospital's rule):

(7.7)

lim a„k

n-1 j t 1 E T~a E &sas,k-1 2o."

n^oo

\T0tH

=

\Tn.Un

—

2 ni

•.•oo

In concluding this paper, we note that the results of Section 4 (bounds on the distribution of T0>„) are obviously valid also for an arbitrary Markov process with a finite number of states for which (7.15)

M exp {

— zt0>„}

=

1

P(z)'

where P(z) is a polynomial, all of whose roots are negative. The equality (7.15) will, for example, be satisfied for a process in which instantaneous transitions upward can only occur by one state at a time. In the general case, where (7.16)

M

exp

{ - z t

0

, „ }

=

Ç ( Z )

P(z)

86

SIXTH BERKELEY SYMPOSIUM:

SOLOVIEV

and the roots of the polynomial P(z) are negative, the asymptotic behavior of the variables T0 „ can be investigated by our methods, if certain restrictions are imposed upon the growth of the polynomial Q(z). REFERENCES [1] D. R. Cox, Renewal Theory, London, Methuen; New York, Wiley, 1962. [2] W. FELLER, An Introduction to Probability Theory and Its Applications, Vol. 1, New York, Wiley, 1968 (3rd ed.). [3] G. M. FICHTENHOLZ, Course in Differential and Integral Calculus, Moscow and Leningrad, Fizmatgiz, 1963. (In Russian.) [4]

B.

V.

GNEDENKO,

YU.

K.

BELYAYEV,

and

A.

D.

SOLOVIEV,

Mathematical

Methods

of

Reliability Theory (edited by Richard Barlow), New York, Academic Press, 1969. [5] S. KARLIN and J. L. MCGREGOR, "The differential equations of birth-and-death processes, and the Stieltjes moment problem," Trans. Amer. Math. Soc., Vol. 85 (1957), pp. 489-546. [6] I. P. NATANSON, Constructive Function Theory (translated by J. R . Schulenberger), New York, Ungar, 1965.

MARTIN BOUNDARIES OF RANDOM WALKS ON LOCALLY COMPACT GROUPS R.

AZENCOTT

U N I V E R S I T Y OF C A L I F O R N I A ,

BERKELEY

and P. I.H.E.S.,

CARTIER BURES-SUR-YVETTE

Introduction Let G be a separable locally compact space and let (Xt), t in T, be a transient Markov process with values in G, where T is either the set of positive integers (discrete time) or the set of positive real numbers (continuous time). Let (Q') be the semigroup of transition kernels of (X,). Let / and X be, respectively, a positive Borel function on G and a positive measure on the Borel cr-field of G. Call / (respectively, X) excessive if Q'f ^ / a n d lim ( _ 0 Q'f = / (respectively, XQ' ^ X), and invariant if Q'f = f (respectively, XQ' = X). Around 1955, the early studies of excessive functions of a Markov process centered around two problems: the relations between Brownian motion and Newtonian potential theory, and the behavior of the trajectories of the process (X,) as t -* + 0 0 . The latter approach can be traced back to D. Blackwell ([4], 1955) who noticed the link between bounded invariant functions and the subsets of G in which (X,) stays, from some finite time on, with positive probability. The importance of these 'sojourn' sets became clear after W. Feller's magistral article ([24], 1956), where they are used to construct (discrete T and G) a compactification G u F of G such that each bounded invariant function / extends continuously to G u F and is uniquely determined by its values on the Feller boundary F. The other approach was initiated by two papers of J. L. Doob : a study of the behavior of subharmonic functions along Brownian paths ([16], 1954), and a probabilistic approach to the potential theory of the heat equation ([17], 1955). The relation between potential theory and general (transient) Markov processes was completely clarified by G. Hunt soon after ([32], 1957-1958). These two trends of thought each found their expression in Doob's work ([18], 1959) which revived the methods used by R. Martin ([39], 1941) in his classical study of harmonic functions. In this article, Doob constructed (for The work of R. Azencott was partially supported by N S F Grant G P - 2 4 4 9 0 . 87

88

S I X T H BERKELEY S Y M P O S I U M : AZENCOTT A N D CARTIER

discrete T and G) a compactification of G by essentially adjoining to G a set of extreme invariant functions, obtained the integral representation of excessive functions in terms of extreme excessive functions, and proved the basic results about the almost sure convergence of (X t ) to the Martin boundary, as t -* +00. Hunt ([33], 1960) introduced new methods of achieving Doob's results, in particular the reversal of the sense of time for Markov processes. It soon became clear that Feller's boundary is almost always too large (D. Kendall [34], 1960, J. Feldman [26], 1962), and in later studies the approach of Doob and Hunt has prevailed. The extension of potential theoretic results to the recurrent case began around 1960-1961, first for the case of random walks (K. Ito and H. J. McKean, and F. Spitzer) and then for the case of Markov chains (J. G. Kemeny and J. L. Snell). A very lucid exposition of the main ideas of boundary and potential theory for Markov chains (for discrete T and G) was given by J. Neveu ([46], 1964), using G. Choquet's results on convex cones ([11], 1956) to obtain the integral representation of excessive functions. In [36] (1965), H. Kunita and T. Watanabe treated the case of continuous time and general state space. They considered two processes in duality with respect to a measure on the state space, a situation whose importance had been recognized earlier by G. A. Hunt [32] and P. A. Meyer (thesis). The construction of the (exit) boundary uses, then, the compact caps of the cone of coexcessive measures (that is, measures which are excessive with respect to the dual process). Meyer ([42], 1968) gave a new presentation of their results using Choquet's integral representation theorem. When the state space G is a topological group, a class of Markov processes is naturally linked to the group structure, the random walks on G. We now restrict our attention to this situation. In the abelian case, the first significant result was reached by G. Choquet and J. Deny ([12], I960), who showed that the extreme invariant functions are essentially characters of the group. For the particular case when G = Z, group of integers, this was noticed simultaneously by J. L. Doob, J. L. Snell, and R. Williamson ([22], I960). For the case G = Z", the boundaries of random walks were then carefully described by P. Hennequin ([31], thesis 1962) and the asymptotic behavior of the Green function at the boundary was obtained by P. Ney and F. Spitzer ([47], 1966). The basic results of the potential theory of random walks on Zn, exposed by Spitzer ([49], 1964) were soon extended to countable abelian groups in a joint paper with H. Kesten ([35], 1966). For the nonabelian case, the extreme invariant functions were obtained first for finitely generated groups by E. Dynkin and M. Maljutov ([23], 1961). A major advance was made by H. Furstenberg ([27], 1963) giving an integral representation of bounded invariant functions for random walks on semisimple connected Lie groups, containing as a particular case the classical Poisson formula relative to harmonic functions on a disc. He obtained partial results ([29], 1965) about the cone of all nonnegative invariant functions for the same

R A N D O M W A L K S ON G R O U P S

89

class of groups. The main results of [27] were extended by R. Azeneott ([3], 1970) to a larger class of groups. We have tried to outline briefly the evolution of the main currents of ideas relevant to our work. Lack of space has forced us to make a number of sizeable omissions: the essential influences of "abstract" potential theory, the important applications of the theory of excessive functions, and the rich material concerning the recurrent case have been barely alluded to. The "Poisson formula" obtained by Purstenberg [27] involves a family of explicitly described, compact homogeneous spaces of 0, the "Poisson spaces" of G. One question arises naturally: what are the relations between the Poisson spaces of G and the Martin boundaries of random walks of G ? This problem is solved in the last part of the present work. We study the case of a transient random walk on a locally compact separable group; there is then a natural random walk in duality with the first one with respect to any right invariant Haar measure on G. Let U be the potential kernel of the dual random walk. To any function r ^ 0 on G such that 0 < Ur < co ("reference" function), we associate as in [36] a continuous one to one map from G into a compact cap of the cone of coexcessive measures. As in Neveu [46] and Meyer [42], Choquet's theorem gives, then, the integral representation of coexcessive measures. The classical, Martin type compactifications of G have been abandoned here, mainly because G does not in general act continuously on these spaces. Even a strong restriction on the type of reference function used ("adapted" reference function, see Sections 16 and 17) only insures an action of G on part of the boundary. The only favorable case seems to be the one when the closed semigroup generated in G by the support of the law of the random walk is large enough (see Section 8). We have preferred to imbed G in the sets of rays of the cone of coexcessive measures obtaining thus a Hausdorff space G and j £ f(x)fi(dx) for the integral of a function / in b + (E) with respect to the measure fi; such an integral may be infinite. By definition of ex, one gets f(x) = for any / in b + (E). The measure /x on E is called a Radon measure if it enjoys the following properties: (a) (local finiteness) every point of E has an open neighborhood V such that /¿( V) is finite; (b) (inner regularity) for every Borel subset A of E, the number fi(A) is the L.U.B. of the numbers ¡x(K) where K runs over the class of compact subsets of A. When n is a Radon measure, fi(K) is finite whenever K is compact and among the closed subsets of E whose complement is fi null there is a smallest one called the support of //. When E is a separable locally compact space, local finiteness means that fi(K) is finite for K compact, and it implies inner regularity [8]. Let E and E' be Hausdorff spaces. A kernel Q from E into E' is a mapping from b + (E') into b + (E) such that Q(Z™= i fh) = 2n°°=1 Qf'n for/; in b + (E'), n ^ 1. The kernel Q is called markovian if and only if Q\ = 1. Let fi be a measure on E\ then there exists a unique measure fiQ on E' such that (nQ,f'y = (fx, Qf'y for each /' in b + (E'). In particular, to Q there corresponds a map q from E into the set of measures on E' given by q(x) = exQ. Then Qf'(x) = / ' ) for /' in b + (E'). If jx is a measure on E, one gets (1.1)

for each Borel subset A' of E'; we shall abbreviate this relation as nQ = JE q(x)fi(dx). Finally, let Q' be a kernel from E' into another Hausdorff space

RANDOM WALKS ON GROUPS

91

The composite kernel QQ' f r o m E into E" is defined by (QQ')f' = Q(Q'f") for / " in b + (E"). By duality, one gets ¡x(QQ') = (fiQ)Q' f o r each measure ¡1 on E. 1.2. Topological groups. Let 0 be a separable locally compact group. The convolution n*fi' of two measures /J. and / / on 6 is the image of the product measure (i gg' f r o m G x G into G. N o t e the following integral f o r m u l a (1.2)

1 and (S„)n^0 as random processes carried by the separable locally compact group G,,. W e shall say that an element g of G is fi recurrent if and only if each neighborhood of gr is hit infinitely often by almost every path of the process ($„)„> 0 - The following theorem is an easy generalization of the results of Chung and Fuchs [15]. According to the previous remarks, there is no real loss of generality in assuming G^ = G and this hypothesis simplifies the enunciation. THEOREM 2.1. Assume that there is no proper closed subgroup of G containing the support of fi. Let us define the measure it = 0 ¡x" on G. There is the following dichotomy: (i) (transient case) no element of G is fi recurrent and nis a Radon measure; (ii) (recurrent case) every element of G is fi recurrent and n( V) is infinite for every nonempty open set V in G. PROOF. Let R be the set of fi recurrent elements. W e shall prove that R is equal to 0 or to G. For that purpose, we introduce the set S of all elements g of G enjoying the following property: for every open neighborhood V of g there exists an integer n ^ 0 such that P[S„ e F ] > 0. The complement of 8 in G is the largest open set U such that P[£„ e U~\ = 0 for all w ^ 0, hence contains no fi recurrent point. It follows that S is closed and contains R. W e prove next the inclusion S~ lR a R. Indeed, let g be in 8 and h be in R, and let U be an open neighborhood of g~1h. By continuity of the operation in G, we can find open neighborhoods V and W, of g and h, respectively, such that V_1W c U. By definition of S, there exists an integer k ^ 0 such that the event A = e 7 ] has positive probability. Let A' be the set of all co in A such that there exist infinitely many integers n ^ 0 such that Sn+k(a>) e W. One gets A' € and since W is an open neighborhood of the ¡x recurrent point h, one has P [ 4 ' ] = P [ J ] > 0. Put S'n = S^Sn + k for n ^ 0; the process (S'n)„^0 is then independent from Zlt • • • , Zk, hence from A', and for every co in A' the relation Sn+k(co) e W entails S'n{co) = 8k(co)~1Sn+k(co) e V~lW