Contributions to Probability Theory [Reprint 2020 ed.] 9780520325340

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PROCEEDINGS FIFTH

BERKELEY

OF

SYMPOSIUM

VOLUME I I

Part 2

THE

PROCEEDINGS of the FIFTH BERKELEY SYMPOSIUM ON MATHEMATICAL STATISTICS AND PROBABILITY Held, at the Statistical Laboratory University of California June 21—July 18,1965 and December 27,1965—January 7,1966 with the support

of

University of California National Science Foundation National Institutes of Health Air Force Office of Scientific Research Army Research Office Office of Naval Research VOLUME II Part 2 CONTRIBUTIONS TO PROBABILITY

EDITED BY L U C I E N AND J E R Z Y

M.

LE

THEORY

CAM

NEYMAN

UNIVERSITY OF CALIFORNIA PRESS BERKELEY AND LOS ANGELES

1967

UNIVERSITY OF CALIFORNIA PRESS BERKELEY AND LOS ANGELES CALIFORNIA

CAMBRIDGE UNIVERSITY LONDON,

PRESS

ENGLAND

COPYRIGHT ©

1967,

BY

THE REGENTS OF THE UNIVERSITY OF 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. LIBRARY OF CONGRESS CATALOG CARD N U M B E R :

49-8189

PRINTED IN THE UNITED STATES OF AMERICA

CONTENTS OF PROCEEDINGS VOLUMES I, II, III, IV, AND V Volume I—Theory of Statistics General Theory T. W. ANDERSON and S. M. SAMUELS, Some inequalities among binomial and Poisson probabilities. R. R. BAHADUR, An optimal property of the likelihood ratio statistic. GEORGE A. BARNARD, The use of the likelihood function in statistical practice. D. BASU, Problems relating to the existence of maximal and minimal elements in some families of statistics (subfields). YU. K. B E L Y A E V , On confidence intervals and sets for various statistical models. F R I E D H E L M E I C K E R , Limit theorems for regressions with unequal and dependent errors. R. H. FARRELL, Weak limits of sequences of Bayes procedures in estimation theory. R. H. FARRELL, J. KIEFER, and A. WALBRAN, Optimum multivariate designs. JAROSLAV HAJEK, On basic concepts of statistics. J. L. HODGES, JR., Efficiency in normal samples and tolerance of extreme values for some estimates of location. J. L. HODGES, JR. and E. L. LEHMANN, Moments of chi and power of t. WASSILY HOEFFDING, On probabilities of large deviations. P E T E R J. HUBER, The behavior of maximum likelihood estimates under nonstandard conditions. OSCAR KEMPTHORNE, The classical problem of inference—goodness of fit. H. KUDO, On partial prior information and the property of parametric sufficiency. YU. K . LINNIK, On the elimination of nuisance parameters in statistical problems. JAMES MACQUEEN, Some methods for classification and analysis of multivariate observations. K A M E O MATUSITA, Classification based on distance in multivariate Gaussian cases. EMANUEL PARZEN, On empirical multiple time series analysis. YU. V. PROHOROV, Some characterization problems in statistics. R O Y R A D N E R , A note on maximal points of convex sets in I C . R. RAO, Least squares theory using an estimated dispersion matrix and its application to measurement of signals. K A R O L Y SARKADI, On testing for normality. R. A. WIJSMAN, Cross-sections of orbits and their application to densities of maximal invariants. Sequential Procedures P E T E R J. B I C K E L and JOSEPH A. Y A H A V , Asymptotically pointwise optimal procedures in sequential analysis. D A V I D B L A C K W E L L , Positive dynamic programming. Y. S. CHOW and H. ROBBINS, A class of optimal stopping problems. Y. S. CHOW and H. ROBBINS, On values associated with a stochastic sequence. A R Y E H D V O R E T Z K Y , Existence and properties of certain optimal stopping rules. THOMAS S. FERGUSON, On discrete evasion games with a two-move information lag. M. V. JOHNS, JR., Two-action compound decision problems. JERZY LOS, Horizon in dynamic programs. Information Theory T. K I T A G A W A , Information science and its connection with statistics. A L F R E D R f i N Y I , On some basic problems of statistics from the point of view of information theory. M. ROSENBLATT-ROTH, Approximations in information theory. J. WOLFOWITZ, Approximation with a fidelity criterion. Nonparametric Procedures P E T E R J. BICKEL, Some contributions to the theory of order statistics. R A L P H B R A D L E Y , Topics in rank-order statistics. Z. GOVINDARAJULU, L. LE CAM, and M. v

vi

CONTENTS OF

PROCEEDINGS

RAGHAVACHARI, Generalizations of theorems of Chernoff and Savage on asymptotic normality of nonparametric test statistics. PRANAB KUMAR SEN, On a class of two sample bivariate nonparametric tests. I. VINCZE, On some questions connected with two sample tests of Smirnov type.

Volume II—Part I—Theory of Probability Probability on Algebraic Structures SIMEON M. BERMAN, Sign-invariant random elements in topological groups. R. GANGOLLI, Abstract harmonic analysis and Levy's Brownian motion of several parameters. LEONARD GROSS, Abstract Wiener spaces. E D I T H MOURIER, Random elements in linear spaces. C. RYLL-NARDZEWSKI, On fixed points of semigroups of endormorphisims of linear spaces. A. and C. IONESCU TULCEA, On the existence of a lifting commuting with the left translations of an arbitrary locally compact group.

Distributions in Functional Spaces HERMANN DINGES, Random shifts of stationary processes. T. HIDA and N. IKEDA, Analysis on Hilbert space with reproducing kernel arising from multiple Wiener integral. K. ITO, Generalized uniform complex measures in the Hilbertian metric space with their application to the Feynman integral. A. V. SKOROHOD, On the densities of probability measures in functional spaces. LESTER DUBINS and DAVID A. FREEDMAN, Random distribution functions.

Stochastic Processes and Prediction HARALD CRAMER, A contribution to the multiplicity theory of stochastic processes. R. M. DUDLEY, On prediction theory for nonstationary sequences. K. URBANIK, Some prediction problems for strictly stationary processes. A. M. YAGLOM, Some topics in the theory of linear extrapolation of stationary random processes.

Martingales M. BRELOT, Capacity and balayage for decreasing sets. LESTER DUBINS and GIDEON SCHWARZ, On extremal martingale distributions. STEVEN OREY, F-processes. VOLKER STRASSEN, Almost sure behavior of sums of independent random variables and martingales.

Special Problems D. A. DARLING, Some limit theorems associated with multinomial trials. GERARD DEBREU, Integration of correspondences. WILLIAM FELLER, On regular variation and local limit theorems. MILOSLAV JIRINA, General branching processes with continuous time parameter. EUGENE LUKACS, On the arithmetical properties of certain entire characteristic functions. HERMAN RUBIN, Supports of convolutions of identical distributions. E. SPARRE ANDERSEN, An algebraic treatment of fluctuations of sums of random variables. LAJOS TAKACS, On combinatorial methods in the theory of stochastic processes.

Volume II, Part II—Theory of Probability Markov Processes R. M. BLUMENTHAL and R. K. GETOOR, Accessible terminal times. LEO BREIMAN, First exit times from a square root boundary. E. B. DYNKIN, General lateral conditions for some diffusion processes. H. KESTEN, The Martin boundary of recurrent random walks on countable groups. M. MOTOO, Application of additive functionals to the boundary problem of Markov processes (Levy system of U-processes). TADASHI UENO,

CONTENTS OF PROCEEDINGS

vii

A survey on the Markov process on the boundary of multidimensional diffusion. H. K U N I T A and T. WATANABE, Some theorems concerning resolvents over locally compact spaces. DAVID G. K E N D A L L , On Markov groups. J . M. O. SPEAKMAN, Some problems relating to Markov groups. D A V I D WILLIAMS, Uniform ergodicity in Markov chains. J. G. B A S T E R F I E L D , On quasi-compact pseudo-resolvents. J . M. O. SPEAKMAN, A note on Markov semigroups which are compact for some b u t not all t > 0. D A N I E L RAY, Some local properties of Markov processes. D O N A L D O R N S T E I N , A limit theorem for independent random variables. C H A R L E S STONE, On local and ratio limit theorems. J O H N L A M P E R T I , Limiting distributions for branching processes. SAMUEL K A R L I N and J A M E S M c G R E G O R , Uniqueness of stationary measures for branching processes and applications. W. L. S M I T H , Some peculiar semi-Markov processes. W. L. S M I T H , A theorem on functions of characteristic functions and its application to some renewal theoretic random walk problems. F . SPITZER, Renewal theorems for Markov chains.

Ergodic Theory R O B E R T J . AUMANN, Random measure preserving transformations. J. R. BLUM, H. D. B R U N K , and D. L. HANSON, Roots of the one-sided N-shift. R. V. CHACON, A geometric construction of measure preserving transformations. A. G. H A J I A N and Y U J I ITO, Conservative positive contractions in ZA K O N R A D JACOBS, On Poincare's recurrence theorem. SHIZUO K A K U T A N I , Ergodic theory of shift transformations. U L R I C H K R E N G E L , Classification of states for operators. KLAUS K R I C K E B E R G , Strong mixing properties of Markov chains with infinite invariant measure. CALVIN C. MOORE, Invariant measures on product spaces. JACQUES N E V E U , Existence of bounded invariant measures in ergodic theory. M U R R A Y ROSENBLATT, Transition probability operators.

Volume III—Physical Sciences Astronomy E. M. B U R B I D G E and G. R . B U R B I D G E , Evolution of galaxies. W. H. McCREA, Age distribution of galaxies. T H O R N T O N PAGE, Masses of galaxies: singles and members of multiple systems. B E V E R L Y LYNDS, Space distribution of small dark nebulae. W. C. LIVINGSTON, On correlations between brightness velocity and magnetic fields in the solar photosphere.

Physics R . L. D O B R U S H I N , Existence of phase transitions in models of a lattice gas. J. M. H A M M E R S L E Y , Harnesses. H E R B E R T SOLOMON, Random packing density.

Spectral Analysis M. S. B A R T L E T T , The spectral analysis of line processes. B E N O I T M A N D E L B R O T , Sporadic random functions and conditional spectral analysis; self-similar examples and limits.

Control Processes J O H N B A T H E R and H E R M A N C H E R N O F F , Sequential decisions in the control of a spaceship. R I C H A R D BELLMAN, On the construction of a mathematical theory of the identification of systems. P . W H I T T L E , The deterministic-stochastic transition in control processes and the use of maximum and integral transforms.

Reliability R I C H A R D E. BARLOW and A L B E R T W. MARSHALL, Bounds on interval probabilities for restricted families of distributions. YU. K . BELYAEV, B. V. G N E D E N K O , and A. D. SOLOVIEV, On some stochastic problems of reliability theory. Z. W. B I R N B A U M

viii

CONTENTS OF PROCEEDINGS

and J. D. ESARY, Some inequalities for reliability functions. B. V. GNEDENKO, Some theorems on standbys. F R A N K PROSCHAN and RONALD PYKE, Tests for monotone failure rate. A. D. SOLOVIEV, Theory of aging elements.

Volume IV—Biology and Problems of Health Information, Processing, and Cognition MARY A. B. BRAZIER, The challenge of biological organization to mathematical description. R I C H A R D BELLMAN, Adaptive processes and intelligent machines. H. J. B R E M E R M A N N , Quantum noise and information. VIOLET R. CANE, Mathematical models for neural networks. E D W A R D A. FEIGENBAUM, Information processing and memory. JULIAN FELDMAN, Recognition of pattern in periodic binary sequences. WALTER R E I T M A N , Modeling the formation and use of concepts, percepts, and rules.

Demography NATHAN KEYFITZ, Estimating the trajectory of a population. M I N D E L C. SHEPS, Uses of stochastic models in the evaluation of population policies. I. Theory and approaches to data analysis. E D W A R D B. P E R R I N , Uses of stochastic models for the evaluation of population policies. II. Extension of the results by computer simulation.

Ecology DOUGLAS G. CHAPMAN, Stochastic models in animal population ecology. E. C. PIELOU, The use of information theory in the study of the diversity of biological populations. J. G. SKELLAM, Seasonal periodicity in theoretical population ecology.

Epidemiology C. C. SPICER, Some empirical studies in epidemiology. D. E. BARTON, F. N. DAVID, EVELYN F I X , M A X I N E M E R R I N G T O N , and P. MUSTACCHI, Tests for space-time interaction and a power function. P. MUSTACCHI, F. N. DAVID, and EVELYN F I X , Three tests for space time interaction: a comparative evaluation. NORMAN T. J. BAILEY, The simulation of stochastic epidemics in two dimensions. R O B E R T BARTOSZYNSKI, Branching processes and the theory of epidemics. J. GANI, On the general stochastic epidemic. H. E. DANIELS, The distribution of the total size of an epidemic.

Genetics THEODOSIUS DOBZHANSKY, Genetic diversity and diversity of environments. HOWARD LEVENE, Genetic diversity and diversity of environment: mathematical aspects. G. MALfiCOT, Identical loci and relationship. OSCAR K E M P T H O R N E , The concept of identity of genes by descent. D. E. BARTON, F. N. DAVID, EVELYN FIX, and M A X I N E M E R R I N G T O N , A review of analysis of karyographs of the human cell in mitosis. J. O. I R W I N , A theory of the association of chromosomes in karyotypes, illustrated by Dr. Patricia Jacobs' data. WALTER F. BODMER, Models for DNA mediated bacterial transformations. SAMUEL KARLIN, JAMES McGREGOR, and WALTER F. BODMER, The rate of production of recombinants between linked genes in finite populations. SAMUEL KARLIN and JAMES McGREGOR, The number of mutants maintained in a population. R. C. LEWONTIN, The genetics of complex systems. P. A. P. MORAN, Unsolved problems in evolutionary theory.

Chance Mechanisms in Living Organisms S. R. B E R N A R D , L. R. SHENTON, and V. R. RAO UPPULURI, Stochastic models for the distribution of radioactive materials in a connected system of compartments. P R E M

CONTENTS

OF

PROCEEDINGS

ix

S. P U R I , A class of stochastic models for response after infection of the absence of defense mechanism. J. GANI, Models for antibody attachment to virus and bacteriophage.

Cellular Phenomena H E R B E R T E. K U B I T S C H E K , Cell generation times: ancestral and internal controls. H. R U B I N , Cell growth as a function of cell density. W A L T E R R . STAHL, Measures of organization in a model of cellular self-reproduction based on Turing machines. D A V I D B U R N E T T - H A L L and W. A. O'N. WAUGH, Sensitivity of a birth process to changes in the generation time distribution.

Carcinogenesis DAVID L I N D E R and S T A N L E Y M. G A R T L E R , Problem of single cell vs. multicell origin of a tumor. WOLFGANG J. B t l H L E R , Single cell vs. multicell hypotheses of tumor formation. T. T I M O T H Y C R O C K E R and B E R Y L J. N I E L S E N , Chemical carcinogens and respiratory epithelium. K . B. DEOME, The mouse mammary tumor system. D A V I D W. WEISS, Immunology of spontaneous tumors. M. B. S H I M K I N , R. W I E D E R , D. MARZI, N. GUBAREFF, and V. SUNTZEFF, Lung tumors in mice receiving different schedules of urethane. M. W H I T E , A. G R E N D O N , and H. B. JONES, Effects of urethane dose and time pattern on tumor formation. J E R Z Y N E Y M A N and E L I Z A B E T H L. SCOTT, Statistical aspect of the problem of carcinogenesis.

Experimentation F. YATES, A fresh look at the basic principles of the design and analysis of experiments. P. A R M I T A G E , Some developments in the theory and practice of sequential medical trials. H E R M A N C H E R N O F F , Sequential models for clinical trials. J E R O M E C O R N F I E L D and SAMUEL W. G R E E N H O U S E , On certain aspects of sequential clinical trials. B R A D L E Y E F R O N , The two-sample problem with censored data. M A R V I N A. S C H N E I D E R M A N , Mouse to man: statistical problems in bringing a drug to clinical trial.

Decision Theory in Medical Diagnosis L E O N A R D R U B I N , M O R R I S F. COLLEN, and G E O R G E E. GOLDMAN, Frequency decision theoretical approach to automated medical diagnosis. CHARLES D. FLAGLE, A decision theoretical comparison of three procedures of screening for a single disease. L E E B. LUSTED, Logical analysis in medical diagnosis. J . T. CHU, Some decision making methods applicable to the medical sciences.

Volume V—Weather Modification Physical Background M. N E I B U R G E R , Physical factors in precipitation processes and their influence on the effectiveness of cloud seeding.

Large Randomized Experiments LOUIS J. B A T T A N and A. R I C H A R D KASSANDER, JR., Summary of results of a randomized cloud seeding project in Arizona. J. B E R N I E R , On the design and evaluation of cloud seeding experiments performed by Electricité de France. W A Y N E L. D E C K E R and PAUL T. S C H I C K E N D A N Z , The evaluation of rainfall records from a five year cloud seeding experiment in Missouri. D O N A L D L. E B E R L Y and L E W I S H. ROBINSON, Design and evaluation of randomized wintertime cloud seeding at high elevation. K. R. GABRIEL, The Israeli artificial rain stimulation experiment statistical evaluation for the period 1961-65. L E W I S O. G R A N T and PAUL W. M I E L K E , JR., A randomized cloud seeding experiment at

X

CONTENTS OF

PROCEEDINGS

Climax, Colorado, 1960-65. E. PÉREZ SILICEO, A brief description of an experiment on artificial stimulation of rain in the Necaxa watershed, Mexico. PAUL SCHMID, On "Grossversuch III," a randomized hail suppression experiment in Switzerland. E. J. SMITH, Cloud seeding experiments in Australia. S. A. CHANGNON, JR. and F. A. HUFF, The effect of natural rainfall variability in verification of rain modification experiments. THOMAS J. HENDERSON, Tracking silver iodide nuclei under orographic influence.

Nonrandomized Operations GLENN W. BRIER, THOMAS H. CARPENTER, and DWIGHT B. KLINE, Some problems in evaluating cloud seeding effects over extensive areas. HANS GERHARD MÜLLER, Weather modification experiments in Bavaria.

Methodological Discussion A R N O l D COURT, Randomized cloud seeding in the United States. L. G. DAVIS and C. L. HOSLER, The design, execution and evaluation of a weather modification experiment. ARCHIE KAHAN, The Bureau of Reclamation's Atmospheric Water Resources Research Program. VUJICA M. YEVDJEVICH, Evaluation of weather modification as expressed in stream-flow response. JERZY NEYMAN and ELIZABETH L. SCOTT, Some outstanding problems relating to rain modification. JERZY NEYMAN and ELIZABETH L. SCOTT, Appendix. Planning an experiment with cloud seeding. JERZY NEYMAN and ELIZABETH L. SCOTT, Note on the Weather Bureau ACN Project. J. M. WELLS and M. A. WELLS, Note on Project SCUD. JERZY NEYMAN and ELIZABETH L. SCOTT, Note on techniques of evaluation of single rain stimulation experiments. ROBERT B. DAVIES and PREM S. PURI, Some techniques of summary evaluations of several independent experiments. BARRY R. JAMES, On Pitman efficiency of some tests of scale for the Gamma distribution. FRANK YATES, Discussion of reports on cloud seeding experiments.

Observational Data A collection of data from cloud seeding experiments in five countries.

PREFACE T H E P U R P O S E OF T H E Berkeley Statistical Symposia, held every five years, is to stimulate research through lectures by carefully selected speakers and through prolonged personal contacts of scholars brought together from distant centers. Accordingly, particular Symposia last from four to seven weeks. On occasion, and this was the case with the Fifth Symposium, they are conducted in two parts, one in June-July, emphasizing theory, and the other in December-January, emphasizing applications. The winter part of the Fifth Symposium was held in conjunction with the 132nd Annual Meeting of the American Association for the Advancement of Science. The Proceedings of the Symposia are intended to present a comprehensive cross-section of contemporary thinking on problems of probability and statistics and on selected fields of application. The rapid growth of research in statistics and especially in probability makes it increasingly difficult to achieve a complete coverage of the field, but sincere efforts are made to invite to the Symposia representatives of all the existing schools of thought, each individual having complete freedom of expression. The organization of the theoretical part of the Fifth Berkeley Symposium was carried out, and the contributors were selected, with the participation of an Advisory Committee composed of Professors J. L. Doob, S. Karlin, and H. Robbins, delegated for this purpose by the American Mathematical Society and by the Institute of Mathematical Statistics. In addition, we had the assistance of Professor D. L. Burkholder, the Editor of the Annals of Mathematical Statistics. The interest of the American Mathematical Society and of the Institute of Mathematical Statistics and their help are deeply appreciated. While a broad coverage of contemporary work in the theory of probability and statistics is difficult, the field of applications of these disciplines is currently so wide that the program of a single symposium can include no more than a few particular domains. The domains covered at the Fifth Symposium were selected on two principles. First, some applied problems appeared as subjects of studies by outstanding probabilists and statisticians invited to the Symposium on account of their work in theory. Second, an effort was made to delineate a few fields of substantive studies that appear particularly promising for probabilistic and statistical treatment. One of the most fruitful fields of this category is undoubtedly biology and problems of health. Here we profited greatly by the advice of Drs. LaMont Cole, Jerome Cornfield, F. N. David, Louis Hellman, Samuel Greenhouse, Hardin Jones, Samuel Karlin, David Krech, Lincoln Moses, Curt Stern, Michael B. Shimkin, and Cornelius Tobias. Quite a few of these colleagues are connected with the broad research activity of the National Institutes of Health and helped to bring to our attention many novel and important subfields of research.

xi

xii

PREFACE

In the field of astronomy we are deeply indebted to Drs. N. U. Mayall, Rudolph Minkowski, and Thornton L. Page. For advice in the field of meteorology we are grateful to Drs. Earl Droessler, James Hughes, Dwight B. Kline, Morris Neiburger, Jerome Spar, Edward P. Todd, and P. H. Wyckoff. Special thanks are due to Dr. Kenneth B. Spengler, Secretary of the American Meteorological Society. Following the established tradition, Volume I of the present Proceedings is given to the theory of statistics. Volume II is devoted to the theory of probability. Because of the large amount of material, about 1000 pages in print, this volume had to be divided into two parts formed through a somewhat arbitrary classification of papers. Volume III includes papers related to physical sciences: astronomy, theory of control, physics, and the theory of reliability. Volume IV, on biology and problems of health, includes papers on information and brain phenomena, on chance mechanisms in live organisms, on epidemiology, on genetics, on medical diagnosis, on clinical trials, on carcinogenesis and cellular phenomena, on demography, and on ecology. Some of these subdomains are already subjects of well developed statistical treatment. Others appear to offer interesting and important possibilities. Compared to the Proceedings of the earlier Symposia, Volume V, being entirely given to the problem of artificial weather modification, is an innovation. With the classification adopted for the first four volumes, weather modification would fit Volume III. It is assigned a special volume because of the specificity of the domain and because of its separateness from all the other fields dealt with in Volume III. Also, the novelty of the problem of weather modification, considered by itself and as a field for statistical research, indicated the desirability of producing a comprehensive coverage of the more extensive experiments. Finally, it appears probable that the readership of the material being published in Volume V will be essentially different from that expected to be interested in Volume III. The fifth Symposium would not have been possible without very substantial financial support from various sources. Hearty thanks are due to Dr. Clark Kerr, President of the University of California, for a special grant made several years in advance of the Symposium. Without this grant, no planning and no initial steps for the organization of the Symposium would have been possible. This initial triggering grant of the University of California was later supplemented by the subsidy of the University Editorial Committee, without which the publication of the Proceedings, to be sold at a reasonable price, would have been a very difficult problem. To a very considerable extent, the theoretical part of the Symposium and the part concerned with physical sciences, were financed by The Program in Mathematics of the National Science Foundation, by the Air Force Office of Scientific Research, by the Army Research Office, and by the Office of Naval Research. The large program on biology and problems of health was made possible by a grant of the National Institutes of Health. The

PREFACE

xiii

program on weather modification was organized using a grant of the Atmospheric Sciences Section of the National Science Foundation. Finally, we wish to record special help from the Office of Naval Research, in the form of air transportation for a number of foreign participants in the Symposium. I t is our pleasure to acknowledge gratefully the generosity of the governmental institutions enumerated. The vitality of our Symposia and the growth of the Proceedings, from 500 pages in 1945 to about 3,000 in 1965, seem to indicate t h a t the funds provided are being spent to fill a real need. The problems connected with the publication of such an amount of technical material are very substantial, especially since some of the material was originally written in languages other than English and required translation. All efforts were made toward speedy publication at a reasonable price, and we are pleased to acknowledge the excellent cooperation and assistance we received from the University of California Press. For the translation of manuscripts, we are indebted to Drs. Amiel Feinstein, Morris Friedman, and Mrs. C. Stein. We are also indebted to several of our colleagues in the Department for work connected with the preparation of manuscripts for the printer. Special thanks are due to Professors E. L. Scott, M. Loeve, E. W. Barankin, to Drs. Carlos-Barbosa Dantas, W. Biihler, Nora Smiriga, Grace Yang, to Mr. Steve Stigler, and Mrs. M. Darland. We are pleased to acknowledge the technical help of Mrs. Sharlee Guise and Mrs. Carol Rule Roth. For taking care of the many complexities of editing technical manuscripts we are deeply indebted to Miss Susan Jenkins whose patience and skill deserve superlative praise. Thanks are also due to Mrs. Virginia Thompson for her greatly appreciated assistance in the same process. To Mr. August Fruge, the Director of the University of California Press, we extend heartfelt thanks for financial, technical, and moral support in publishing so much difficult material. Special thanks are due also to Joel Walters, Editor of the University of California Press. In spite of all our efforts, we found ourselves unable to keep up with the schedule of publication proposed by the Press, but we must thank them for helping us to keep the delays at a minimum and for producing a publication in accordance with the usual excellent standards of the University of California Press. Many thanks are due to our Administrative Assistant, Miss M. Genelly for taking care of many financial and organizational difficulties. For transportation, housing, and other logistic problems connected with the organization of the meeting itself, very valuable assistance was received from the staff of the Laboratory and in particular from Miss June Haynes and Mrs. J. Lovasich. As was the case on many earlier similar occasions, for supervising and taking care of the innumerable intricacies of local organization we are deeply indebted to our colleague Professor Elizabeth L. Scott. It is a pleasure to express here our deepest appreciation.

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PREFACE

Last but not least we wish to thank the Department of Statistics of the University of California, Berkeley, and all our colleagues therein, for their sympathetic attitude and help. Particular thanks are due to David Blackwell. During the winter part of the Fifth Symposium, the Statistical Laboratory lost one of its organizers as well as one of its most active members. Our colleague and cordial friend, Professor Evelyn Fix died of a heart attack on December 30, only a few hours after she acted as one of the hostesses at the banquet of the Symposium. Sit ei terra levis! L U C I E N L E CAM

JERZY NEYMAN

Director, Statistical Laboratory April, 1967

CONTENTS Markov Processes R.

M.

BLUMENTHAL

and

R.

K.

GETOOR—Accessible Terminal

Times

1

LEO BREIMAN—First Exit Times from a Square Root Boundary

9

E. B. DYNKIN—General Lateral Conditions for Some Diffusion Processes

17

KESTEN—The Martin Boundary of Recurrent Random Walks on Countable Groups

51

M. MOTOO—Application of Additive Functionals to the Boundary Problem of Markov Processes (Levy's System of U-processes)

75

UENO—A Survey on the Markov Process on the Boundary of Multi-dimensional Diffusion

Ill

and T. WATANABE—Some Theorems Concerning Resolvents over Locally Compact Spaces

131

H.

TADASHI

H . KTJNITA

G. KENDALL—On Markov Groups

DAVID

M. Groups

JANE

DAVID J. G.

O.

165

SPEAKMAN—Some Problems Relating to Markov

175

WILLIAMS—Uniform Ergodicity in Markov Chains .

BASTERFIELD—On Quasi-compact Pseudo-resolvents .

187 .

193

M. O . SPEAKMAN—A Note on Markov Semigroups Which Are Compact for Some But Not All t > 0

197

JANE

DANIEL DONALD

RAY—Some Local Properties of Markov Processes . ORNSTEIN—A Limit Theorem for Independent Random

Variables

201

213 .

217

LAMPERTI—Limiting Distributions for Branching Processes

225

and J A M E S MCGREGOR—Uniqueness of Stationary Measures for Branching Processes and Applications . .

243

CHARLES JOHN

STONE—On Local and Ratio Limit Theorems .

SAMUEL K A R L I N

xv

xvi

CONTENTS

W. L. SMITH—Some Peculiar Semi-Markov Processes

.

.

255

W. L. SMITH—A Theorem on Functions of Characteristic Functions and Its Application to Some Renewal Theoretic Random Walk Problems 265 F. SPITZER—Renewal Theorems for Markov Chains .

Ergodic

.311

Theory

J. AUMANN—Random Measure Preserving Transforma-

ROBERT

tions

321

J . R . B L U M , H . D . B R U N K , a n d D . L . H A N S O N — R o o t s of

the

One-sided JV-shift R. Y. CHACON—A Geometric Construction of Measure Preserving Transformations A R S H A G B . H A J I A N and Y U J I ITO—Conservative Positive Contractions in L1 JACOBS—On Poincare's Recurrence Theorem

KONRAD SHIZUO

.

KRENGEL—Classification of States for Operators

335 361 375

KAKUTANI—Ergodic Theory of Shift Transformations .

ULRICH

327

.

KRICKEBERG—Strong Mixing Properties of Markov Chains with Infinite Invariant Measure

405 415

KLAUS

CALVIN C . JACQUES

MOORE—Invariant Measures on Product Spaces

.

447

NEVEU—Existence of Bounded Invariant Measures in

Ergodic Theory MURRAY

431

ROSENBLATT—Transition Probability Operators

461 473

ACCESSIBLE TERMINAL TIMES R. M. BLUMENTHAL and R. K. GETOOR UNIVERSITY OF WASHINGTON

1. Introduction Let X = (Q, 3TC, Px, Xh 6t) be a Hunt process having a locally compact space E with a countable base as state space. We refer the reader to the expository paper ([4] or [1], pp. 133-134), for all concepts and notations which are not explicitly mentioned in the present paper. A stopping time T for the process X is called accessible if for each initial measure n on E there is a nondecreasing sequence {T„} of stopping times such that P" almost surely, Tn T and Tn < T for all n on {T > 0}. Meyer [7] has proved the remarkable result that a stopping time T is accessible if and only if the path t —> Xt(co) is continuous at T(u) almost surely on {T < »}. We will say that a stopping time T is thin if PX(T > 0) = 1 for all x in E. As usual, an analytic subset A of E is thin if PX(TA > 0) = 1 for all x in E, where TA = inf {£ > 0: Xt e A} is the hitting time of A. These definitions are consistent since clearly A is thin if and only if TA is thin. Finally a stopping time T is called a terminal time if for each t (1.1)

T = t+

T °6h

almost surely on

{T > t}.

If A is an analytic subset of E, then TA is a terminal time and the phrase "almost surely" may even be dropped from statement (1.1). Let us now assume that X satisfies Hunt's hypothesis (F). (See [5], [6], or [1], pp. 133-134.) It then follows from proposition 18.5 of [5] that TA is an accessible terminal time whenever A is a thin analytic subset of E. Moreover, it is clear that TA = 00 on {TA > f} if A C E. The main result of this paper is the following converse of the above statement. THEOREM 1. Assume X satisfies hypothesis (F). If T is a thin accessible terminal time with the property that PX\X < Z1 < = 0 for all x, then there exists a thin Borel set B C E such that T = TB almost surely. The proof of theorem 1 is given in section 2; then in section 3 we give some applications of theorem 1 to the structure of natural additive functionals. Consider the following process: the state space E = L U Li U L2 is the following subset of the Euclidean plane, L = {(#, y): x < 0, y = 0} is the nonpositive rr-axis, Li is the segment joining the points (0, 1) and (1,0), whereas L2 is the segment joining (0, —1) and (1,0). The process consists of translation to the right at unit speed until (0, 0) is reached. The point (0, 0) is a holding point This work was partially supported by the National Science Foundation, NSF-GP 3781.

1

2

FIFTH B E R K E L E Y

S Y M P O S I U M : B L U M E N T H A L AND GETOOR

with parameter 1 from which the process jumps to (0, 1) or (0, — 1) with probability respectively. The process then moves with unit speed along the appropriate segment L\ or L 2 until it reaches (1,0) where it remains forever. Define T by T(w) = «> if the trajectory t —> X((co) reaches (1, 0) via the lower segment L 2 or if X0(w) = (1,0); whereas, if the trajectory arrives at (1, 0) via the upper segment L1; let T{co) be the time at which the process reaches (1, 0). It is immediate that T is a thin accessible terminal time, and it is equally clear that if the initial measure ¡J. attaches positive mass to L, then there is no thin set B such that T = TB almost surely P", even if we allow the set B to depend on n. This example is, of course, artificial, but it does show that theorem 1 is not valid for Hunt processes in general. One can construct examples which are less artificial. 2. Proof of theorem 1 In the rest of this paper we will assume that X satisfies Hunt's hypothesis (F). We will break up the proof of theorem 1 into several lemmas. In this section (for typographical convenience) we will write U and PB, rather than UL and PB, for the potential kernel and hitting distributions obtained by taking the auxiliary parameter X to be 1; that is, for any bounded Borel measurable /, / U(x, y)f(y) dy = E* /q" e-'/(X,) dt,

(21)

PBKX) =

E*{e-T°F{XTB)}.

Here, as in Hunt, £(dy) = dy denotes the basic measure on E. We will also need the fact that if T satisfies the hypotheses of theorem 1 and R is any stopping time, then R + T ° ds = T almost surely on {R < T}. This follows easily from the strong Markov property for multiplicative functionals [6]. From now on T will always satisfy the hypotheses of theorem 1. Let (x) = Ex(erT). Since T is a thin terminal time it is easy to see that is 1-excessive and that is strictly less than 1. According to theorem 18.7 of [5], we may write = Un + where ¡± is a measure on E and ^ is a 1-excessive function with the property that PFt = whenever F is the complement of a compact subset of E. The following notation will be used in the remainder of this section. Let Kn ={> 1 — 1 /n}. Each Kn is a finely closed Borel set, and the Kn are decreasing with empty intersection. Let Tn = TK„ be the hitting time of Kn. LEMMA 1. For each n, PKJ> = , and almost surely T„ | T with Tn < T on {T< co}. PROOF. Fix an x and let {R,} be an increasing sequence of stopping times such that Px(Rn —> T, Rn < T for all n) = 1. Such a sequence exists, since T is a thin accessible time. Now (2.2) But

RN

E'ie-^Xa.)} | T,

= ^ { e x p (-Rn

- T » 6U}

and hence lim„ } since is 1-exces-

3

ACCESSIBLE TERMINAL TIMES x

sive, must equal one on {T < co}, all of these statements holding P almost surely. Recalling the definition of Kn, it is clear that Tn < T almost surely Px. Moreover, on {Tn = T < } = {Tn = T < one has for each m > n that {XR) = 4>(XTJ > 1 — (1/m), and this contradicts the fact that is strictly less than one. Consequently, Tn < T almost surely Px on {T < oo}. In particular this yields (2.3)

= Ex{exp ( - Tn - T . 6T.)} = Ex{e~T) = 4>(x).

PkMX)

Finally, the relationship (2.4)

E'{e-^~T.);

Tn
(XTJ; T . < (1 - ^

Px(Tn

< oo)

implies that Tn f T almost surely Px. Since x is arbitrary, this completes the proof of lemma 1. Define Ln = {x: x is left regular for Kn} and let B be the intersection of the Ln- Each Ln is a countable intersection of open sets because Ln — {x: £x(e-T„) = l} ; and excessive functions are lower semicontinuous. I t will turn out that B is the set we are looking for; that is, T = TB almost surely. LEMMA 2. The measure n is carried by B. PROOF. Since Pk„u < u for any 1-excessive function u and PkA = , it follows that PKnUix = Uji and PkJP = 4 oo. I t is not difficult to conclude from this that Up. is a potential of class (D) (see [4] or [6] for the definition), and consequently, according to Meyer's result [6], there exists a unique natural additive functional A of X such that Un(x) = Ex J? e~l dA(t) for all x. LEMMA 3. Let R = inf {t: A (t) > 0} ; then R = T almost surely. PROOF.

(2.5)

W e have

U^x)

= PKM^(x)

= Ex [

e-< dA(t),

and hence A(T„) = 0 almost surely. However, Tn | T, and this implies that T < R almost surely. We turn now to the opposite inequality. As a first step, we will show that \(/(XT„) ^ ( X T ) almost surely on {T < oo}. By assumption, T = oo almost surely on {f < T}, hence it suffices to prove the convergence on {T < f}. Let {D„} be an increasing sequence of compact subsets of E whose union is E. If Sn is the hitting time of the complement of Dn, then S„ f f as n —> oo. For a fixed fc define Qn — min (T n , Sk) and Q — min ( T , Sk). Clearly Q„ f Q, and it suffices to show that ^(XQJ 'P(XQ) on {Q < oo}. If x is fixed, {e-cV(X QN) hence, if A is in 5QM, then for all N > M (2.8)

A} > EX{E^(XQ);

E*{E-H{XQ.))

A},

and letting N —> E x {e~mX Q ); A}.

(2.9)

Using the characterization of JFq given in [2], it is immediate that (2.9) holds for all A in SQ, and H and E~Q\P{XQ) being "SQ measurable, it follows that H > E~QYP(XQ) almost surely PX. In view of (2.7) and the definition of H, this implies that \I/(XQ) almost surely on {Q < Thus we have shown that ^(XT„) —»*P(XT) almost surely on {T < M; then we have (2.10)

EX{E~T"4>{XT^);

A} = EX{E~TNUFI(X?.);

A} + E

X

{ E ^ ( X

T

J ; A},

with a similar expression in which T replaces TN. Since $(XT„) —» TP(XT) and (XTJ —* 1 almost surely on {T < co}, by letting N—> and then by subtracting the corresponding expression involving T, one obtains (2.11)

E*{E~T[

1

-

4>(XT)]',

A} =

EX{E~TA{T)\

A}.

It now follows that this must hold for all A in JFr, and consequently, (2.12)

A(T)

= 1 - (XT) > 0

almost surely on {T < » } . But this implies that R < T almost surely, and so lemma 3 is established. LEMMA 4 .

THE INEQUALITY T > TB HOLDS ALMOST SURELY.

By construction, UN(X) Meyer [6] (see also [1]) implies that PROOF.

(2.13)

f

U{X, Y)KY)N(DY)

= EX J "

e~'dA(t), and hence a result of

= E* F~

E~'F{XT)

DAT

for all bounded Borel measurable /. Taking / to be the characteristic function of E\B and using lemmas 2 and 3, one finds (2.14)

0 =

EX ¡{T

M)

E-'KXT)

DAIT) >

EX{E^F(XT)A(T)}.

But A(T) > 0 on {T < • °°. Of course, BN = KN for each N.

A C C E S S I B L E TERMINAL LEMMA 5 .

Prg>PBg.

Let v be a positive measure on E such that g = Uv is bounded, then

Recalling the definition of all x and fL, < TB. Consequently, PROOF.

(2.15)

5

TIMES

PKnPBmg

=

Ln and B, one has PX[TL„ < f Kr]

UPK,PBmv

>

UPbPBmV

=

= 0

for

PBPBmg-

Let Rn,m = Tn + TBm ° 0t„, then (2.16)

Pk„PbMx)

= ^{cxp

(-Rn,m)g[X(Rn,m)]}.

We now claim that for ra, fixed R„,m coincides with T + TBm ° 6t for sufficiently large n almost surely on {T < °o } . To prove this we note that since Bm is finely open, we have

Px[Tn + TBn ° 6Tn < T + TBm . 6T; T < «] < P*[Xt e Bm for some t e [Tn, T); T < »]. However Tn^T and 4>{Xt) 1 as t | T on {T < « } , and hence this last

(2.17)

expression approaches zero as n —>• oo. Therefore, Rn,m = T + TBm » dT for sufficiently large n almost surely on {T < oo}. It now follows from (2.16) that Px„PBmg —> PrPBmg as n - > « for each fixed m, and hence PtPBJ3 > PbPb^But PBmg increases to j as m -> ®, and this establishes lemma 5. We may now easily complete the proof of theorem 1. We have just shown that Ex{e~Tg(XT)} dominates Ex{e~TBg(XTB)} whenever g is a bounded potential Uv. But the function identically equal to one on E is the limit of an increasing sequence of such potentials, hence Ex(e~T) > Ex(e~TB). Combining this with lemma 4 completes the proof of theorem 1. REMARK 1. Naturally the set B is finely closed since it is thin. In addition, the particular set B constructed above is cofinely closed in as much as each Ln is cofinely closed, that is, closed in the fine topology for the dual process It. 3. Natural additive functionals In this section U and Pb will have their usual meanings; that is, they are the potential kernel and the hitting distributions for X = 0. Let A = A (i) be a natural additive functional of X, and for simplicity, we assume that A has a finite potential, that is, u(x) = EX{A (=o)} < oo. (The following results are valid with the obvious modifications if one only assumes that A has a finite X-potential for some strictly positive X.) In [6], Meyer has shown that A can be decomposed into the sum of a continuous additive functional, C, and a purely discontinuous natural additive functional, D, in the following manner. For a given n let T(n} (to) = T(w) be the smallest value of t such that |w[X((«)] — •u[X(_(o))]| > 1 /n and the path X(is continuous at t. Here u(XtJ) denotes lim3 f t u(Xs) and not w(lim s f ( Xs). I t is known that T satisfies the hypotheses of theorem 1 for each n (see [6] or [3]). Let Gn(«) be the magnitude of the jump at T, that is

6 (3.1)

F I F T H B E R K E L E Y SYMPOSIUM '. B L U M E N T H A L AND GETOOR

(?„(«) = u[X(T( w ) - , « ) ] - u[X(T(co), «)] > 0.

It is clear that G„ = 0 on {T = oo} and Gn > l / n on {T < We now define the successive jumping times T'^ = Tk by Ta = 0 and Tk+1 = Tk + T ° for fc > 0. Next define the additive functional Dn(t) by (3.2)

Dn(t, «) = L G„(0rtco),

the sum being taken over those k for which Tk(«) < It is easy to see that D„ is a natural additive functional for each n, and Meyer has shown that limB Dn (t) exists uniformly on [0, oo) almost surely and defines a natural additive functional D(t). Finally the difference u(x) — EX{D(oo)} is the potential of a continuous additive functional C(t), and consequently A(t) = C(t) + D(t). This decomposition is valid for general Hunt processes and so does not depend on hypothesis (F). From now on we assume that (F) holds; then for each n there exists by theorem 1 a thin set Bn such that T(n> = TB„ almost surely. Let Bd be the union of the Bn, so that Bd is semipolar, and let Bc = E^Bd- Moreover, the T fn) are decreasing, and therefore we may assume the Bn are increasing. THEOREM 2 . Let Id and Ic be the indicator functions of Bd and Be respectively; then (3.3)

D(t) = Jol Id(Xu) dA(u)

and

C(t) = j* IC(XU) dA(u),

where, as usual, the equality of additive functionals means equivalence. PROOF. If R is an accessible terminal time, a standard argument, like the one used in the proof of lemma 3, shows that (3.4)

u(XRJ) - u{XR) = A(R) — A(R—)

almost surely on {R < oo }. Therefore, it follows that T(n) is the first t such that Ait) — A(t—) > l / n and that Gn is the jump in A at that point. If Tin> is finite, X(T(n)) is in Bn almost surely since Bn is thin. Consequently, we may write (3.5)

Dn{t) = JT' IBSXU)

dA(u),

and letting n—»oo we obtain the assertion about D(t). The one about C(t) is then obvious since I c + I d = 1. REMARK. If the only semipolar sets are polar, then it is an immediate consequence of theorem 2 that the only natural additive functionals (with finite potential) are continuous. One can find simple examples to show that this is not the case if we assume only Hunt's hypothesis (A). We will close this section with one more comment on the structure of D(t). Fix n and consider the approximating functional Dn(t) and also the natural additive functional «/„( 0 the left side of (3.8) vanishes, then using the relationship between J n and Dn, one sees that / Ul(x, y)f(y)nn(dy) also vanishes. But this, together with the uniqueness of potentials, implies that ¡xn is absolutely continuous with respect to vn. Letting dnn = gn dvn, we have (3.9)

Dn(t)

=

£

gn(Xu)

dJn(u).

In particular, G„ = gn[X(TM)] almost surely, and off Bn. Moreover, if m > n, we may write (3.10)

Jn(t)

=

|q' IbSXu)

gn

may be assumed to vanish

dJm(t),

because A (i) jumps by more than l / n at time TJbm) if and only if X(Tim)) is in Bn. Hence vn is the restriction of vm to the set B„. Thus we may assume that gn and gm agree on Bn; that is, we may define a nonnegative Borel measurable function g on E vanishing off Bd such that g„ is the restriction of g to Bn for each n. Now we may write the approximating functionals Z)„ as (3.11)

Dn(t) = Z

g[xm]

k

where the sum is over those k satisfying < t. Thus the purely discontinuous additive functional D is completely determined by the function g and the increasing sequence of thin sets {Bn}. Finally if one defines Ai = Bi, An±i — Bn+i — U"=i Aj and the times R'kn} to be the successive hitting times of An, that is, Ri,n) = 0 and (3.12)

R f U = RLn) + Ta„ • dR(n>,

for

k > 0,

then one has the following representation of D : (3.13)

Dit)

=

1 1 n

k

g i x m i

where again the inner summation is over those k satisfying R™ < t. One can show by simple examples that the jump G„ will not always be expressible as a function of the position X(TM) alone, if one merely assumes Hunt's hypothesis (A). In particular, the representations (3.11) and (3.13) are not valid for arbitrary Hunt processes.

8

FIFTH BERKELEY

SYMPOSIUM: BLTTMENTHAL AND GETOOR

REFERENCES [1] R. M. BLUMENTHAL and R. K. GETOOR, "Additive functional of Markov processes in duality," Trans. Amer. Math. Soc., Vol. 112 (1964), pp. 131-163. [2] , "A theorem on stopping times," Ann. Math. Statist., Vol. 35 (1964), pp. 1348-1350. [3] R. K. GETOOK, "Additive functional of a Markov process," Lecture Notes, University of Hamburg, 1964. [4] , "Additive functionals and excessive functions," Ann. Math. Statist., Vol. 36 (1965), pp. 409-422. [5] G. A. HUNT, "Markoff processes and potentials I I I , " Illinois J. Math., Vol. 2 (1958), pp. 151-213.

[6] P. A. P. MEYER, "Fonctionelles multiplicatives et additives de Markov," Ann. Inst. Fourier {Grenoble), Vol. 12 (1962), pp. 125-230. [7] , "Decomposition of supermartingales: the uniqueness theorem," Illinois J. Math., Vol. 7 (1963), pp. 1-17.

FIRST EXIT TIMES FROM A SQUARE ROOT BOUNDARY LEO

BREIMAN

UNIVERSITY OF CALIFORNIA, LOS ANGELES

1. Introduction The thing that motivated the present paper was a curious observation by Blackwell and Freedman [1], Let Xh X2, • • • , be independent ± 1 with probability i , Sn = Xi + • • • + Xn, and Tc = min {n; Sn > cVn, cVn > 1}. Then for all c < 1, ETC < «>, but c > 1 implies ETC = In order to understand this better, I wanted to calculate the asymptotic form of P(TC > n) for large n. It was reasonable to conjecture that P(TC > n) cm-", not only for cointossing random variables but for a large class. The first step in the proof of this was to verify the result for Brownian motion. This is done in the second paragraph and follows easily from known results. To go anywhere from there, one would like to invoke an invariance principle. But the difficulty is clear—for general identically distributed, independent r.v. Xi, X2, • • • with EXi = 0, EXi = 1, the most one could hope for is that P(TC > n) ~ an~P where /3 is the same for all distributions, but a depends intimately on the structure of the process. Hence, this is not a situation in which the usual invariance principle is applicable. But the result does hold for all distributions such that E\Xi\3 < «>, and it is proved by using results of Prohorov [2] which give estimates of the rate of convergence of the relevant invariance theorem. This proof is carried out in the third paragraph. A dividend of the preceding proof is collected. The conclusion is that (1.1)

P(Sn
1}, that is, T* is the first exit time past t = 1. Preparation of this paper was sponsored in part by National Science Foundation Grant GP-1606.

9

10

F I F T H B E R K E L E Y SYMPOSIUM:

THEOREM

1.

With the above definition,

BREIMAN

P(T*C > i|£(l) = 0) ~ atr^

where

(i) lim 0(c) = 0, C—»«

(ii) lim /3(c) = oo, (iii) if c2 is the smallest positive root of ™ (-2c-)n ml n =o (2n) ! (m — n)\

(2.1)

then ¡3(c) = m. In particular /3(1) = 1, /3(V3 - V 6 ) = 2. PROOF. Consider the process Y(u) = £(e2u)/e". This is the Uhlenbeck process. The problem now is to find the distribution of the first exit time Ty for Y(u) from the boundaries ± c , given F(0) = 0. This distribution is well known, and its Laplace transform has been given by Bellman and Harris [6] and Darling and Siegert [7]. To wit; (2.2)

3>(X) = J^

e~xv dP (Ty > V) =

+ D-x(0) £>_x(c) + D_x(—c)

1

2*/2r(X/2)

2 Jo" e-d/2)(^-i cosh ci dt

R(X >

0,

where the D\(z) are parabolic cylinder functions. Now Z)_x(c) is an entire function of X (see Erd^lyi et al. [8], pp. 117 f.f.), so $(X) is entire except for poles on the nonpositive axis. Since D-\(z) + D-\(—z) satisfies the self-adjoint SturmLiouville equation (2.3)

+ (J - i z 2 ) * = \,

it follows, under the supplementary condition $(a) = ( — a), that the characteristic values of this system coincide with the zeroes of D-\(a) + Z)_x(—a). Therefore, the poles of $(X) are simple and real. Let —2/3(c) be the position of the largest pole. Then (2.4)

P(TY

>V)

= AE-W + 0(e~ (2 " +s>v ),

For the Brownian motion, this translates as (2.5)

P(T% >t)

= atraw + 0(£~~"-(8/2)).

Now to verify (i), (ii), and (iii) by locating the largest zero of (2.6)

0(X) =

Ol-X/2 1

f"

^ / e - ^ x - i cosh ct dt. W*) Jo

To continue the integral into Rt\ < 0, write

5 > 0.

SQUARE ROOT BOUNDARY

JQ e - ^ x - i

cosh ctdt =

(2.7)

jf

+ r

e(X)

e-tV2tx-1

27

11

|^CoSh a -

2V2-! f n=o(2n)!

£

( § y j ) ] dt + Mi)) r(X/2)

_ 21-x/2 - , , f c*» r(n + (X/2)) " f p ) /Ar(X ' C) + . ? „ (2n)l r(X/2)

The function In{\, C) is analytic for RFX > —2N — 2, real and positive for real X in this range. For X = —2m, the finite sum becomes (2.8)

P(c) = t ( " 2 C 2 ) " ,=o (2 n)\

(m-n)\

m!

and the first term in (2.7) vanishes. This gives (iii). As c —» 0, 0(A) —» 1 for all X, so for any number M, and for c sufficiently small, 0(A) can have no zeroes in |X| < M. For (i), use N = 0 in (2.3) to get (2.9)

r Q ) 0(X) = 2i-w»/0(X) + r Q ) .

RtX > - 2 .

Since 70(X)—»0 as c — t h e root must move toward a pole of T(X/2). But this can only be at X = 0. 3. First exit distribution for sums of independent r.v. Let Xi, Xi, • • • be independent, identically distributed r.v. EX\ = 1, EIX^ < oo. Define Tc = min {n, Sn > cVn}, Sn = Xt THEOREM 2. Either there exists an integer n such that P(TC P(TC > n) ~ an~PM, where /3(c) is the same function appearing in a > 0 will not, in general, be the same. PROOF. (3.1)

with EXx = 0, + • • • + Xn. > n) = 0, or theorem 1, but

The proof is constructed around the use of the identity Pn(y) = j Qn.Uy, r,)Pm(dv),

m < n,

where (3.2) (3.3)

Pn(y) = P (J^L < Qn,m( y, v) = P i-^r Wn

Y>

< y> r%l

»),

= m, ••• ,n\ -^2= = A vm /

Take m = [e2u] (the largest integer < e2u) and n = [e 2( "+ M) ], then the invariance principle results in

12

FIFTH BERKELEY SYMPOSIUM: BREIMAN

(3.4)

lim Qn,m(y, v) = P(Y(uo)

71—* 60

= Q(y,

< y, | F ( F ) | < c, 0 < 7 < w„|F(0) = r,)

v)

(see [7]). However, what is needed is a uniform error bound. PROPOSITION 1.

There

(3.5)

is a constant

D such

that

sup |Q,. b ( 7> v) - Q(y, v)\


0.

Write

PROOF.

(3.6) p ( J k . W n


for y either variable y or 77. For w0 —• 0, J h(y)Q(dy, 17) —> h(y) for reasonable h(7), hence (3.22)

Q t ( 7 , „) = e-hr'/a) I ] k = 0

where 1^(2/), A* satisfy the eigenvalue equation, (3"23)

% -

y

t y

=

X+>

*(c) = * ( - c ) = 0.

Write (J,g) = i±cce~W2)f(y)g(y)dy, then I(y, s) for the second and third terms. Therefore, (3.24)

e - f Uy)P(dy,

s) =

f My)P(dy,

and for a sufficiently smooth function h(y) = (3.25)

j f(y)P(dy,

s) = ± ^

= hj.

s) + f My)I(dy, (h,

^

In (3.20) write

if>

s),

^)^(t), / ** — °°, then s0 must be one of the values X0, Xi, • • • , say Ay, P(IC, s) has a single pole at s = s0, and is otherwise regular in the half-plane Rls > So — B, 8 > 0. (The poles at s0 ± 2n TT i/u0, n ^ 0, are ruled out because P(IC, s) does not depend on Uo.) The statement is also true for j f(y)P(dy, s) for any smooth / such that (/, \j/,) 0. For such smooth /, one can write (3.28)

J f(y)P(dy,

u) = L(f)e^

+ 0(e^-««).

Letting P*(dy, u) = e~XiUP(dy, u), one obtains (3.29)

j f(y)P*(dy,

u) = L(f)

+ 0(e~*>).

The measures P*(dy, u) converge weakly to P*(dy),

where {\pk, dP*) = 0,

SQUARE ROOT BOUNDARY

15

ly2/2)

k j. The latter implies P*(dy) = e~ \p,(y) dy. However, P*(dy) is a nonnegative measure. The only nonnegative \f/j(y) is ^0(7); thus j = 0. Finally, the transformation (x) = e~ ixi/i)\p(x) changes (3.23) into the system (2.3) so that X0 = —2/3(c). The only thing remaining is to verify that so > — if there is no n such that P(TC > n) = 0. For any u0, take I such that for u > uh (3.30)

inf

Q(1,77,

u, Mo) > 5 > 0.

From (3.18), P(I, u + u0) > SP(I, u), u > ui. This will imply that either there is an M > 0 such that P(I, u) > e~Mu, all u > 0, or that there is a u2 such that P(I, u) = 0, u > u2. The first alternative is impossible because of the assumed analyticity of P(IC, s). Put another way, there is an interval J and an m such that P{Sm G J, Tc > m) = 0. Then argue that J can certainly be taken large enough such that P(X 1 e J — x) > 0, all x e J. Therefore, (3.31)

P{Sm e

Tc > m) = 0 =» P(S m _! e J, Tc > m - 1) = 0.

By reduction, an n is arrived at with P(TC > n) = 0. Note that n is not necessarily one by considering X\ to take on the two values 1 /M and M 2 , for M large. The smallest n satisfying P(TC > n) = 0 is the largest n satisfying n/M < cVn. In the course of this proof we have incidentally proven the following theorem. THEOREM 3 . If there exists no n such that P(TC > n) = 0 , then (3.32)

P(Sn < yVn\Tc > n) ^ 9 J^

e~W2

where 6 is a normalizing constant. 4. Remarks Some unresolved problems that are left in this area are concerned with what happens to the tail of the Tc distribution as the boundaries are moved out. More specifically, let Tc{r) = min {n; Sn > cVn + r}. By invoking the usual invariance principle, it is easy to show that (4.1)

lim P(Tc(T) > H) = P(T*C > EO

where T* is the first exit time for Brownian motion past t = 1. This is not as interesting as asking the following questions. (i) By theorem 2, P{Tc(T) > H) ~ a(r)i-« c ) . As r 00 show that a{r) converges to the corresponding constant for Brownian motion. (ii) A bit more strongly, is it true that P(Tc(r) > rt)/P(T*(T) > rt)-> I uniformly as T , t —»00 ? The condition E|Xi|3 < » can be easily weakened down to E\Xi\2+i < 00 and the same methods will work. I suspect that theorem 2 may even be true under only E\X\\l < but that would require better tools. What happens with more general boundaries, for instance, £1/3, or ^/l 4>(t),

16

FIFTH BERKELEY SYMPOSIUM: BREIMAN

w h e r e 4>(t) d o e s n o t increase t o o fast? T h e results for t h e c V i boundaries rely v e r y h e a v i l y o n t h e f a c t t h a t these transform into c o n s t a n t boundaries for t h e U h l e n b a c h process. H e n c e t h e simple m e t h o d s u s e d here d o n o t generalize.

REFERENCES [1] D. BLACKWELL and D. FREEEMAN, "A remark on the coin-tossing game," Ann. Math. Statist., Vol. 35 (1964), pp. 1345-1347. [2] YU. V. PROHOROV, "Convergence of random processes and limit theorems in probability theory," Teor. Verojatnost. i Primenen., Vol. 1 (1956), pp. 177-238. [3] VOLKER STRASSEN, "Almost sure behavior of sums of independent random variables," Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Berkeley and Los Angeles, University of California Press, 1966, Vol. II, Part I, pp. 315-344. [4] D. A. DARLING and P. ERDOS, "A limit theorem for the maximum of normalized sums of independent random variables," Duke Math. J., Vol. 23 (1956), pp. 143-155. [5] Y . S. CHOW, HERBERT ROBBINS, a n d HENRY TEICHER, " M o m e n t s of r a n d o m l y s t o p p e d

sums," Ann. Math. Statist., Vol. 36 (1965), pp. 789-799. [6] R. BELLMAN and T. HARRIS, "Recurrence times for the Ehrenfest model," Pacific J. Math., Vol. 1 (1951), pp. 179-193. [7] D. A. DARLING and A. J. F. SEIGERT, "The first passage problem for a continuous Markov process," Ann. Math. Statist., Vol. 24 (1953), pp. 624-639. [8] A. ERDISLYI et al., The Higher Transcendental Functions, Vol. 1, New York, McGraw-Hill, 1953. [9] E. C. TITCHMARSH, Eigen Function Expansions, Oxford, Oxford University Press, 1946.

GENERAL LATERAL CONDITIONS FOR SOME DIFFUSION PROCESSES E. B. D Y N K I N Moscow UNIVERSITY 1. Formulation of the problem and fundamental results 1.1. Let E be a plane domain bounded by a smooth contour L, and let v(z) be a smoothly varying vector field on L. Let the point y e L be called exclusive if the projection of the vector v{z) on the inner normal to L changes sign at the point y. Let us say that the function u(z) satisfies the boundary condition a if, at each nonexclusive point z of the contour L, the derivative of u in the direction v(z) is zero. We are interested in solutions of the heat conduction equation ( d u t ( z ) / d t ) = Aut(z) in the domain E, which satisfy the initial condition Uo(z) = f(z) and the boundary condition Q. More accurately, our problem is to describe the general form of the lateral conditions at exclusive points, which will, together with the initial and boundary conditions, define a unique solution ut(z) of the heat conduction equation, wherein: (a) ut(z) > 0 i f f ( z ) > 0; (b) ||M(|| < ll/H (we understand ||/|| to be sup \f(z)\ in the union E* of the domain E and the set of all nonexclusive points of the contour L). (An analogous problem for the system of differential equations of Kolmogorov which describes Markov processes with countable phase space was studied by W. Feller [4], However, Feller considered only a special class of supplementary conditions corresponding to "continuous exit" from the boundary. The supplementary conditions we found cover the most general case.) In terms of probability theory the problem may be stated as follows. The heat conduction equation, together with the boundary condition a , prescribes a Brownian motion process in the domain E with reflection from the boundary in the domain E. The behavior of the trajectories after hitting an exclusive point of the boundary is not determined here. The problem is to describe all possible kinds of such behavior. I t is more convenient to pose and solve the problem in the terminology of semigroups of linear operators. Let 8 be some set and ® some c-algebra of subsets of S. Let B = B(8) the space of all bounded ©-measurable functions on 8 with the norm ||/|| = sup |/(z)|. The family of linear operators Th (t > 0), operating in the space B and satisfying the following conditions: (l.l.A) T t / > 0, i f / > 0, (l.l.B) HZVII < l l / H , (l.l.C) T,Tt = Ts+t for any s, t > 0, is called a Markov semigroup in the space 8. 17

18

FIFTH BERKELEY SYMPOSIUM: DYNKIN

The semigroup Tt in the space E* defined by the formula Ttf(x) = ut(x) corresponds to the boundary value problem described above for the heat conduction equation. (The c-algebra of all Borel sets is always considered as the basic / , if fn(x) -^f(x) for all x e S and the sequence of norms ||/„[| is bounded. Let us now define the operator 21 as follows. Let D be the set of all functions from B(E*) having Holder-continuous first partial derivatives in E* and Holdercontinuous second partial derivatives in E, and satisfying the boundary condition a . Let us consider the Laplace operator A in the domain 2D. I t will be proved that a minimum w-closed extension exists for this operator. We denote this extension also by 21. Our purpose is to describe all 2l-semigroups. 1.2. Let us move along the contour L passing the exclusive point y in the direction of the vector 11(7) and at the same time, observing the projection of the vector v(z) on the inner normal to the contour L at the point z. Let us put 7 6 r+, if this projection changes sign from plus to minus, and 7 e r_, if the sign changes from minus to plus. Let us set r = r + U L (this is the set of all exclusive points). I t is expedient to "split" each point 7 e r into two points 7+ and y~. T h e union of all such pairs is denoted by II. T h e decomposition of II into 11+ and II_ corresponds to the decomposition of T into T+ and If F is a function in E*, then F(y+), F(y~) are its limits when z tends to 7 along the contour L from the positive and negative sides, respectively. I t is proved that if F e Da, then the limiting values F(y+), F(y~) exist for all 7 e T. T o each a e TI_ there corresponds just one bounded harmonic function pa{z) satisfying the boundary condition a and such that pa(a) = 1 and pa(fi) = 0 for /3 e I I + , /3 ¿¿a. (If, say, a = 7 + , then pjz) is the probability that the trajectory issuing from z will approach 7 having touched L on the positive side of 7.) 1.3. Let us suppose the following are given. (1) The partition of the set n + into classes. The set of these classes is denoted by 0. (2) For each co e fi there is a set of nonnegative constants c„, cru, (7 e T+), and a measure vw in the space 8 = E* U n _ U 0. Let E* denote the set of all points z of the set E* for which p(z, r + ) > e (the distance between the point z and the set M is denoted by p(z, M)); Xy.c is a function equal to |z — 7I 2 for \z — y\ < t and zero for \z — 7] > e. Let t = p(z, L). Let us put « e fl' if (tw = 0, bw,y = 0 for all 7 e r + and y„(8) < Let us assume that for every co e 0 the following conditions are satisfied: (1.3.A) Vu(E*) < 00 if e > 0,

DIFFUSION

19

PROCESSES

(1.3.B) (t, va) < °o where t = t(z) = p(z, L), and the integral of the function / in measure v (in the whole space S) is denoted by (/, v). (1.3.C) For any y g r + and any sufficiently small e > 0, and (1.1)

Jf| z - y < e \z - y\*Vu{di) < for a i u;

00 ,

(1.3.D) (p a , v^ < (1.3.E) v u ( u ) = 0; (1.3.F) ba,y = 0 if at least one of the points 7+, y~ does not belong to co; (1.3.G) = 0 for co e 0'; (1.3.H) at least one of the numbers ^(S), K,y, (y G r + ) , cu, a; (1.3.b) to each GO e Q there corresponds a number (we call it F(co)) such that F(a) = F(u) for all a G co; (1.3.c) if a«, > 0, there exists an analogously defined value SlF(co); (1.3.d) if bu,,y > 0, there exists d.2)

dn

=

110

t

(1.3.e) for each a i G f i the function F — F(u) is summable in ^-measure; (1.3.f) for each to g fi, (1.3)

(F - F(co),

+

dF E (7) yer+~°" ' dn

cwF(co) -

aJiF(co) = 0.

We say that the 21-scmigroup Tt satisfies the lateral condition 11, if Dg C 3(11). The lateral condition 11 is called a special one if au = 0 for all co G fi. 1.4. The fundamental results of this paper are formulated in theorems 1.11.3. (These results have been published without proof in [3].) Theorem 1.1. Every 31-semigroup satisfies some special lateral condition 11. The arbitrary special lateral condition 11 uniquely determines some 51-semigroup. Theorem 1.1 solves the problem posed in section 1.1. However, the natural question arises of what is the sense of the conditions 11 when some ) in (1.3.b) must be understood, for co e 0, to be the value of the function F at the point w. (Let us note that for w e S2 condition (1.3.f) is automatically satisfied by virtue of the definition of the operator $.) THEOREM 1.2. To an arbitrary set 11 = {cw, 0 where X is a nonnegative constant; (2. l.B) lim i n f ^ F(z) > 0 for all y e r + . Then the function F is nonnegative. When X = 0, and condition (2.1.A) is satisfied with the equality sign, this theorem was proved in ([2], section 5). Additions to this proof, required in the general case, are presented in appendix A. Among the lemmas on which the proof of theorem 2.1 relies, is one which is of independent interest for the sequel. It is the following. LEMMA 2.1. Let a function F, not a constant, be given in the domain E bounded by the smooth contour L, and let \F(z) — AF(z) > 0 for all z e E and for a nonnegative constant X. Let z0 e L and (2.1) F(z0) < F(z) for all z G E.

If F has a derivative in the direction of some vector v making an acute angle with the inner normal at the point z0, then this derivative is -positive. (Here and henceforth, the value F(z0) of the function F at the boundarypoint z0 is the limit of F{z) when z tends to z0 along the set E.) 2.2. In appendix B, a function g(z, w) = gw(z), (w e E, z e E U L, w ^ z) is constructed under the assumption that the set F + is nonempty such that for each w e E: (2.2.A) gw{z) = — (^7r) In \z — w\ + hw(z), where hw(z) is a harmonic function in E) (2.2.B) gw(z) satisfies the boundary condition ft; (2.2.C) gw(z) is bounded in a neighborhood of each point y e T; (2.2.D) gw(y) = 0 for 7 e T+. From the minimum principle (theorem 2.1) it follows at once that the conditions (2.2.A)-(2.2.D) define the function gw(z) uniquely. We shall call it the Green function. It is proved in appendix B that the function gw(z) is nonnegative and has the following property: (2.2.E) the function |z — w\gm(z) is bounded in the domain z e E U L, w e E. 2.3. The functions py(z), (7 e T + , z e E), defined by the conditions: (2.3.A) py(z) is a harmonic function in E; (2.3.B) py satisfies the boundary condition ft; (2.3.C) py is bounded; (2.3.D) py(y) = 1; pyQ3) = 0 for 0 e r+, 0 9* y, play an important part in the construction of the Green function. From the minimum principle it follows that these conditions define the function py uniquely, and that 0 < py(z) < 1. See appendix B for the construction of the function py. The same appendix also gives a proof of the following property: (2.3.E) py(z) = L„er_a}n(z) + Hy(z)

22

FIFTH BERKELEY SYMPOSIUM: DYNKIN

where 0. As has already been remarked in section 2.4, to do this it is sufficient to verify that (2.2) has only a trivial solution for X > 0 and / = 0. For this verification we shall use properties (2.5.A)-(2.5.G) forX = 0. If F + \GF = 0, then according to (2.5.C), F e C°. By virtue of (2.10), (2.5.G), and (2.5.B), F e 3D, \F - AF = 0 and F = 0 on r+. By the minimum principle, F = 0. 2.6. We shall now prove several new properties of the Green operator. (2.6.A) For every X > 0, the general form of the function i 1 e 3D is given by the formula (2.11)

F = G\f + AX|

( / e 0; (2.6.C) \G\1 0 for z e E. We shall denote a point with coordinates s, t by z(s, t). Let us put zs = z(s, 0) = yeia. Let d(s) denote an angle which the vector v{zs) forms with the positive direction of the contour L at the point s. Let us note that 8(0) = 0 and 6 changes sign from plus to minus at the point 0. Hence, k — — 0'(0) > 0. I t is easy to see that oo, [£((1/X)AF)| < ||(1/X)AF| - + 0 . Hence, from equality (5.1) we obtain in the limit (5.2)

F(co) =

UF).

According to the remark at the end of section 5.1, (5.3)

4(F) =

(F, Hu)

34

FIFTH B E R K E L E Y

SYMPOSIUM:

DYNKIN

where /i„ is a finite measure on the space 8. We have (5.4)

(1,

= 4 ( 1 ) < 1.

From (5.2) and (5.3) we have (5.5)

F(co) = (F,

5.3. Let us put co e Ui if fiu is a unit measure concentrated at the point co, and let fl0 = fi\Qi. For co e Oi, equation (5.5) becomes an identity which all the functions F satisfy. In this case, another passage to the limit is necessary. Let us note that for co e fii, (5.6)

lim

tii) =

U / ) = (/,

= /(«),

(/ e

P)

(the limit is taken over some sequence of values of X which tend to + » ) . Let us put

fp0 =

- £ F(t)pt, [Ea ¡P = Fo+E AF(X)GVi.

(5.7)

F

ren

Evidently,

AF = AF - Z

(5.8) (5.8)

re«

From (5.1), (5.7), and (5.8), we have (5.9)

t ( F 0 ) + £ £ ( p r ) [ F ( r ) - F(«)] - [1 - £ ( l ) ] f ( « ) A fea

When X —» (5.10)

f)

^(p r )[AF(T) - A F(«)] - ^

A

AP(co) = 0.

along the sequence selected earlier, then according to (5.6),

tt(F0)->0,

t(AF)^0,

£(pr)-»0,

for

f ^ co, £ ( 1 )

1.

The function F 0 belongs to the space P defined in 3.6. By virtue of (3.41), Fo e Pi. According to the remark at the end of section 3.6, the functional It induces some linear functional on the space Pi. Let nt denote the norm of this induced functional. Let us put Oa = 0\ {co}, (5.11)

5l = n£ + Z t{V() + \ + 1 ren

A

t(l).

For any / e Ph |^(/)| < n£||/||p,. The space Pi is separable. Hence, linear functional may be constructed in Pi, and a sequence of values X may be selected which converges to + iw(f) for all / e Pi. Passing to a subsequence, if necessary, one can satisfy relations (5.6) and (5.10), and at the same time insure the existence of the limits (5.12)

lim^-g.,,«-*«); 0U

lim 1 ~

0U

=

cM;

l i m ^ = 0) are continuously differentiable in the neighborhood of the point y and for f ^ w, (5.15)

$„.r = qH,t - (pr, K) ~

0 . on PROOF 1. The representation (5.14) of the functional 4 results from lemma 3.4. Let us show that (1.3.F) is satisfied. Let a be that one of the two points y+, y~ which does not belong to co, and let f be a class from O, containing a. By virtue of (5.12) there exists a constant c such that for all X of the sequence under consideration (5.16)

t(pt)

£

7 er+

K

< cSl

According to theorem 3.2, for every N > 0 there exists e > 0 such that (5.17)

pa(z) > Nry(z)

for

|z - y\ < e.

Let us consider a function \//(z) given in E U L which satisfies the inequalities 0 < ^ < 1 everywhere, is zero for \z — > 2e, and one for \z — y\ < e. Evidently for all z e E U L, p^z) > pa(z) > NTy(z)^(z), and hence (5.18)

t(Pi)

>

Nt{r^).

Let us note that r y $ e Pi. Hence (5.19)

l i m % ^ = L(r^) = ( r ^ , ?„) + bw,y. 0ai From (5.16), (5.18), and (5.19), we have ba 0. According to (4.6), p}(z) is represented as the sum of functions pa(z), (a e f). Since f either does not contain any of the points y~, y+ or contains both, and since the functions p«, (a e n + , a ^ y~, a 7 + ) and py = py- + py+ are continuously differentiable in the neighborhood of y; this is also true for the function p Let us put 7 G r m if ba,y > 0, and let us consider the continuous function fn(z) in E*, which equals p{(z) for p(z, Ta) < 1/n, equals zero for p(z, T a ) > 2/n and is everywhere between zero and one. The function fn coincides with P( near rM and equals zero near r + \ r w (for sufficiently large n). Hence, /„ e Pi and

36

FIFTH BERKELEY SYMPOSIUM: DYNKIN

(5.20)

- Ufn)

But &(V{) > t(jn). (5.21)

qa,r

= (fn, ?„) + E bu,y

(7).

Therefore,

= lim

> U f n ) = ( / » , ¿G +

E bu.y ^ y

on

(T).

Since /„ —> for n —», then (5.15) results from (5.21). 5.5. The expression (5.7) for the function F0 may be rewritten as follows: (5.22)

E - F(«)]p r . rea Substituting this expression into (5.13) and taking into account lemma 5.2, we have (5.23)

F0 = F - F(u) -

(F - F(u), ?.) +

E

7 er +

K,y 2 (T) + E « k r [ f ( f ) - f («)] on fen - CMF(W) - 0. PROOF. According to (6.3)-(6.4), lemma 2 of appendix C is applicable to the matrix (a£, f ). In order to prove lemma 6.1, it is sufficient to verify that the set K described in lemma 2 is empty. We know that if w e K, the equality sign holds in (6.4) and au,{ = 0 for co e K, f £ K. Hence, the equality v ( t i \ K ) = 0 follows, as does (6.5). I t is clear that K C 0'. But according to (1.3.G), yM(0') = 0. Hence, v ( K ) = 0. However, v „ ( K ) = v { Q \ K ) = 0 together with (6.5) contradict (1.3.H). 6.2. According to section 1.4, we put S = E* U where f2 is the set of all w e Q, for which 0. We shall also use the notation B, 21, 5(11) introduced rl.i

a

a

a

W

in section 1.4. We shall write/„ —> / if f j | fn 11 is bounded. For each / e B we put (6.6)

H l ( J )

( 6 . 7 )

Q l { f )

= ((?x/, =

£

re«

v.)

n

{ z ) — * f ( z )

+

«) + E

for all

b

w

z

e S and the sequence

. y { B * , f } ,

7

r l j H U f ) ,

where r£,r are defined in lemma 6.1. In the space B let us consider the operators R\ defined by the formula R i J

( 6 . 8 )

THEOREM 6 . 1 .

For

any

X >

=

(?x/

0

the

+

Z

operator

Q t ( f ) p l

R \

maps

B

in

a

one-to-one

way

5(11). The inverse mapping is given by the operator X3 — 21. PROOF. According to sections 4.1 and (3.2.C), the general form of the functions satisfying conditions (1.3.a)-(1.3.b) is given by onto

DIFFUSION

(6.9)

39

PROCESSES

F = GJ + E

Qlpl

All these functions automatically satisfy conditions (1.3.d) and (1.3.e). Let F e 3(11). According to the above, F has the form (6.9). According to sections 1.4 and 4.1, (6.10)

\F{z) - tF(z)

= \F(z) - %F(z) = f(z)

for 2 e E*.

The values of /(w) remain undetermined as yet for « e 0. Let us put /(01) = \F(o>) — $LF(o>). Let us recall that f(P(co) is defined by (1.4). Let us now note that the function F defined by (6.9) satisfies condition (1.3.f) if and only if the constants Qt satisfy the system of equations (6.11)

E

=

Hl(f).

By virtue of lemma 6.1, (6.11) is equivalent to (6.7). Hence, the condition F € 3(11) is equivalent to the condition F — R\f, (/ e B). From the relation (A3 — %)R\f = f already proved, the remaining statements of the theorem result. 6.3. Condition (1.3) takes the form (1.6) for w e W. We may rewrite it as (6.12)

F(u) = {F, i>„)

where 5>u = ((v„)/(l, v„) + c). Evidently, (?„, 1) < 1. Let P(1l) denote the set of all functions F e Pa satisfying the conditions (6.12) for all co e Q'. Let us put F e 35(11) if F e 3(11), and 21F e P(cU.). I t is clear that 35(11) C P(1l). There results from theorem 6.1 that for any X > 0 (6.13)

35(11) = i?x[P(1l)].

Our purpose is to prove the following theorem. THEOREM 6.2. The set SD(ll) is everywhere dense in P(1l) (in the sense of uniform convergence). Let us first prove some auxiliary propositions: (6.3.A) P ( 11) is everywhere dense in B (in the sense of w-convergence); (6.3.B) if/„ then - Rrf || 0; (6.3.C) the strong closures of the sets 3)(1l) and R\(B) coincide. PROOF OF (6.3.A). Let 2e, and satisfying the inequalities 0 < B < 1 everywhere• For any -point 7 from Tc a function Cy(z) may be constructed such that Cy(7) = 1 and Cy(z) = 0 at all points of r c except 7. PROOF. Let 6(s) denote the angle between v(eis) and the positive direction of L at the point e". On the segment [0, 1] let us construct a twice continuously differentiable function b(r) equal to 1 near 1, equal to zero near zero, and such that 0 < b(r) < 1 for all r. A function A(z) may be given by the formula

The functions Bt and Cy are obtained by means of the same formula. In order to obtain Bt, it is possible to start from the function a, which equals 1 for \z ~ T[ < 5«, equals zero for \z — y\ > t, and satisfies the inequality 0 < a < 1 at all the rest of the points of the contour L. The function bir) must be selected so that it equals zero for r < 1 — In order to determine Cy, it is sufficient to construct the function a{z) on the contour L so that it equals zero in the neighborhood of the set r c \{7} and satisfies the equality a(eu) = 1 + {' tan 0(s) ds J 80 for s0 — « < s < s0 + e (if 7 = exp (is0)). LEMMA 6 . 3 . If for all Holder-continuous functions f (6.17)

(6.18)

E ky[Gf(7+) ?er-

- Gf{7-)] = 0,

then all the constants ky are zero. PBOOF. Let fn(z) be Holder-continuous functions in E U L such that: U(z) = 0 for \z — w\ > (1 /n), and {/„, 1} = 1. Relying on the minimum principle, it is easy to show that the functions Gf„ converge to g(z, w) uniformly in the neighborhood of Hence, from (6.18) there results (6.19) £ my[g( y+, w) - g(y~, w)] = 0 , {we E). To conclude, apply theorem 1 of appendix B.

41

DIFFUSION PROCESSES PROOF OF THEOREM 6.2.

By

virtue

of

the

Hahn-Banach

theorem

and

(6.3.C), it is sufficient to prove that every linear functional I on the space Pa which vanishes on R\(B), will vanish also on P(11). According to lemma 3.3 and the remark of section 5.1, ((F) = (F, £), where £ is a signed measure on the space £. Thus, let (6.20) (Rxf, £) = 0 for all / e B. I t is necessary to prove that (F, £) = 0 for all F g P(cU). 1. Let us put (6.21) a . = G&{); rt=Zti.tQ»By virtue of (6.6)-(6.8), the relation (6.20) is equivalent to the relation

E W O " ) + £ rtbt.y G (7) = 0 tea refi,7Gr+ on where F = G\f, v = £ + Erea riv(- Let 2) denote the set of all functions F g 3D, which equal zero on I+. According to (2.6.A), every function F from £> may be written in the form G\f, (/ G B), where f = \F — AF. Hence, for any function F G i) the following corollary of equality (6.22) is satisfied: (6.22)

(F, v) +

(6.23)

(F, v) + Z rv&m ~ AF(f)] + E rfo.y ^ (7) = 0. r f,7 on 2. Let us prove that ru = 0 for all u g Let bu,y > 0. Let us consider the function B(, constructed in lemma 6.2. It is easy to see that for sufficiently small e > 0 the function Ft = Bt(l — pa) belongs to £). For this function the relation (6.23) becomes (6.24)

(F„ v) - rj>„.y &

(y) = 0.

Since (dpu/dn)y 0, and (F e , v) —*• 0 as e —»• 0, then ra = 0. Analogously, considering the function Ft = BtG\ 1, we arrive at the relation T{ 0. It is now seen from (6.23) that (F, v) = 0 for all F g £>. Since £> contains all smooth functions which equal zero near L, then v is concentrated on S\E. Considering the functions A(z) and Cy(z) constructed in lemma 6.2, we conclude that v is concentrated on i2 U n_, where f(7 + ) + v(y~) = 0 for all 7 G r_. Hence, the validity of the conditions of lemma 6.3 results from the equality (Gf, v) = 0 (for ky = v(y+)). From lemma 6.3, it follows that ky = 0, (7 e r_). This means the measure v is concentrated on i l Since the set O is finite, the measure v is also finite. Therefore, °o > (pu> v) = (P (see (1.3.D)). Hence, if rw ^ 0, then (pu, vw) < » and therefore, the measure vw is finite and 0 in the formulations and proofs of lemmas 5.15.4 of [2]. Hence, the proof of lemma 5.1 does not change. The coincidence of the exact lower bounds of the function h on the two sets is stated in each of the lemmas 5.2-5.4. In our case, these statements remain valid under the additional assumption that, each time, at least one of the two lower bounds under

45

DIFFUSION PROCESSES

consideration is negative. Hence, in the proofs of lemmas 5.2 and 5.3 it is necessary to consider the auxiliary function (2)

3C e (z) =

h{z)

and the function (3)

H(z)

= h(z) - u - ù

1

+ u

in the proof of lemma 5.4, where k = min (k, 0). After these modifications, theorem 2.1 is derived from lemmas 5.1-5.2 exactly as theorem 5.1 has been derived in [2], 0

0

APPENDIX B.

0

0

0

The Functions py(z) and gw(z)

1. Functions py(z), (y e T + ) satisfying conditions (2.3.A)-(2.3.D), have been constructed in [2] (see theorem 5.2 and section 5.7). The property (2.3.E) results from formulas of section 4.8 of [2], The function gw(z) satisfying conditions (2.2.A)-(2.2.D) has been constructed in section 6 of [2]. This function is nonnegative. It is determined by the formulas (1)

2irgw(z)

(2)

qw(z)

= qw(z)

-

= Re

£ yer+

qw(y)py(z),

Gw{z)

dz.

The form of the function Gw{z) depends on the sign of the index t of the vector field v{z). Let us first assume that ( > 0. Then for w ^ 0, (3)

Gw(z)

=

«•)«,-«-!

i

w — z

-

c l

e-^w*

+

|

+ Z \ w* — zj

where w* = w-1 and a{z) is an analytic function in the circle E, which has a Holder-continuous derivative a'{z) in the closed circle E \J L (see section 4.4 of [2]). This formula is not suitable for studying gw{z) for values of w near zero. Let us put (4)

f

Gw{z) = er*M \

Z)lff(w)

Lw — z

z2tw*

p-iff(w)

^

w* — zj

1

I t is easy to verify that the difference f(z) = Gw(z) — Qw(z) is regular in the circle E for any w ^ 0, continuous on E*, bounded in E U L, and satisfies the relation Re {f{z)ei"| — In \z0 — w\. JZQ

I t is easy to verify that the functions Rw{z) and dRw(z)/dz are continuous and, therefore, bounded in the domain 2 e E U L, w e E". The functions Rw(z) and dRw{z)/dz are continuous in the domain z e E \J L, w G E" and the estimates 3C2 < 3Ci + « *, 2 — w*\

(10)

dRw(z) dz

^ 3C3 5C4 ~ \z — w\ Iz — V)

are satisfied for them (3C< are constants dependent on p). From the relations (11)

dVuj -7 ax

. SVuj T-I % -r— = /£«, ay

dvw -r ax

. drw TJ 4 - r - = Rw ay

there results that the functions (drw/dx), (drw/dy), (d2rw/dx2), {d2fw/dxdy), 2 (d^rv/dy ) are continuous in the domain 2 e E U L, w e E"; the functions (drw/dx), (drw/dy), (»

e) + F c (. Similarly, (1.17) follows from (2.10). The proof of (1.7) and PA — A = S as given in [16] or ([17], propositions 1.3 and 13.3) need only trivial changes. Finally, when this is applied to the reversed r.w., one obtains P*A* — A* = Ô which is the same as AP — A = 5. The proof of theorem 1 is therefore complete.

60

FIFTH BERKELEY SYMPOSIUM: KESTEN

We end this section with the following remark. REMARK 1. If {xn} Q © converges to a boundary point, then (2.38)

lim HB(xn, y) exists

N—>»

for all finite sets B C © and y e B. The proof is the same as in theorem 30.1 of [17]. One takes limits as ?i —» j0. If © is an infinite cyclic group, then (3.4) holds without the condition that has one boundary point only. PROOF. If {Xn} has one boundary point only, then with C — {e, c}, (3.5)

lim Hc(x, e)

X—>«0

exists (see (1.24)), and it follows from (2.29) and (2.19) that c must have the same value as (3.5) (see [13], beginning of proof of theorem 3.1). Equation (3.3) follows, therefore, from (1.17) by taking limits (|x| —>«

On the other hand, for x ^ e, e,y e B C (3.7)

g(x, y) > HB(x,

y) +

E

0(x, t)P(t,

tGB-e

y).

In fact, the left-hand side represents the expected number of visits to y, up to and including the first visit to e when the r.w. starts at x. In the right-hand side of (3.7), HB(X, y) is the expected number of visits to y at the first entrance to B, and E g(x, t)P(t, y) represents the expected number of visits to y, coming in one step from a point of B, but without having visited e. Because e e B, both terms in the right-hand side count only visits up to and including the first visit to e. If one takes limits as |x| —» » in (3.7) and uses (3.6), (1.7), and (1.8), one obtains (3.8)

lim sup

H

b

(X,

y) < 6(e, y) + A{e, -

y) £

f£B

A(e, t)P{t,

y) =

E

t&B

Me,

t)P(t,

y).

Since A > 0 and, again by (1.8), EOO

lim sup mj, < °o i—>»

for each fixed n, then

(3.19)

lim H§(x, y) = 0,

y e

X—*K

In particular,

= « and Iq < ©, or [©:£]


, e)] = 0

for y e !q and w e ® . Thus, for fixed ui, • • • , uk, 1 k •>00 S»v 7" = 1 If § < © and |£>| . Then, using (2.13), (3.21)

lim H$(hzi, y) = lim ^ Z HQiziUr, y).

(3.22)

Z E^hziUr, y)= £ y) = £ «) = 1, r=l «£§ u£® since h,yefQ and ^ is a normal subgroup. For the second part of the lemma, we take Uj = gmi. Equation (3.20) remains valid, even though u, now depends on i, because of (3.18). Instead of (3.22), we now obtain from (3.16) and (3.17), (3.23)

£ H§(hZigmi, 7y) < £ r—1 uE$

u) < 1.

MARTIN BOUNDARY

63

If |§| < =0, then x—>00 if and only if i(x) — s o that (3.15) follows from (3.21) and (3.22). To prove (3.19) under the assumptions (3.16)-(3.18), one only needs to observe that for any k, by (3.21)-(3.23), (3.24)

lim sup II&(hzl, y) < i

—

K

whereas for each fixed i, (3.25) lim H^QiZi, y) = lim H(z{, hrly) = 0. Finally, if = oo but [ © : § ] < then there are only finitely many possible values for z,- and (3.18) is vacuous, but (3.19) follows from (3.25). If = °° and § «

for all

ye £

and such that { F n } , the imbedded r.w. on has only one boundary point, then the r.w. on © has only one boundary point. (Note that lemma 4 is useful in checking condition (3.27). Some applications appear in section 4.) PROOF. A S remarked in the introduction, we have to show that for every ce@-e, (3.28)

lim Hc{x, e) exists,

X—>»

whereas before C — {e, c}. Let § satisfy the assumptions of the theorem and let c e ® — e be fixed. Pick a y0 e § such that H§(c, y0) > 0. Just as in (3.11), one has (3.29)

Hn>(x, y0) > H$Uc{x, c)H$(c, y0),

and thus, by (3.27), (3.30)

lim HWc(x,

X—

c) = 0.

Choose an increasing sequence of finite sets B j C § such that § = U "= i Bj. Again as in (3.11), (3.31)

HSi(x, ya) > HBiuc(x, c)HBi(c, y0) > HBiuc(x, c)H§(c, y0),

if

y0 e Bj.

By lemma 3, applied to {F„}, we can for any given e > 0 find a j0 such that e, i/o e Bj„ and (3.32)

lim

F B , ( i , y0) < eH^c, y0),

64

FIFTH BERKELEY SYMPOSIUM:

KESTEN

and thus by (3.31) and (2.3), (3.33)

lim sup HBhUc(t,

c) < e.

Consequently, there is a ji such t h a t for j > ji, (3.34)

HBiouc(t, c) < 2e

for

te £ -

B,.

Now (3.30) states t h a t when the starting point x is far out, Xn will hit § before c with a probability close to one. By (3.27), the point where § is entered first will be outside B j with a probability close to one, and by (3.26), Xn will then enter Bj, before c except for a set of probability at most 2e. Formally, this argument runs as follows. For some |0 ji and x sufficiently far out, the last three terms of (3.35) are together a t most 3e (by (3.30), (3.27), and (3.34)). Since {Y n } has one boundary point only, (3.36)

lim HbJU, u—• «0 ,u

Z) =

lim HBjQ(u, Z) exists, u—• oo ,u

(see remark 1 and (2.3)). One can therefore choose j > ji so large t h a t (3.37)

| E

HbS, —

z

)Hc{z, e) lim

£

HBk(u, z)Hc(z,

e)| < e

for

t 0 was arbitrary, (3.28) follows. THEOREM 4. The r.w. {Xn} on & has two boundary points if and only if there exists an infinite cyclic subgroup § = (c), such that < oo and cr 2 (F) < 2 where o- (F) is the variance of the imbedded r.w. {F„} on § (see (2.1) and 2.2)).

65

MARTIN BOUNDARY

Note. If a subgroup § as specified exists, but with cr2(F) = then {F„} has only one boundary point by (2.5), and then {X„} has only one boundary point by theorem 3 and lemma 4. Thus the condition a 2 (F) < °o turns out to be independent of the choice of PROOF. First assume that there exists an = (c) as specified in the theorem. Let z0 = e, zi, • • • , be representatives of its cosets where X = [ © : § ] < oo if and only if \k(x) \ —> co, and by remark 2, (3.40)

lim |E B {cH h e) - HB(ck, e)\ = 0

for B = {e, 6}, b e ® — e fixed. Thus, if we show that for each b e ® — e, (3.41)

l i m # B ( c * , e) h—xe

and

lim HB(c~k,e) k-*— °°

exist, then { J „ } has at most two boundary points. Since for 71—»»

0.

These cases clearly exhaust the situations in which the boundary may have more than two points. It therefore suffices to show that in these cases { ! „ } has infinitely many boundary points.

66

FIFTH B E R K E L E Y

Case (1).

SYMPOSIUM'.

KESTEN

Choosefc0such that with B = {e, ch'},

(3.47)

lim 1lB(ck, e) ^ Ic—> 00 ft—*

lim TlB(ck, e). — 00

Such a k0 exists since { F „ } has two boundary points (.see (2.6) and (2.7)). Define (i) CXN{X) = Px [first entrance of { X „ } to § is at ck with k >

N],

(ii) PN(X) = Px [first entrance of { Z „ } to § is at & with k < —A7]. Let {x n } C ® be a sequence for which (3.48)

a = lim

and

—

n

/3 = lim ftv(:r„) —

n

exist and such that (see (3.2) for notation) (3.49)

(n —> oo).

i(xn)-+°°,

By (3.44), these limits are independent of N and a + ¡3 = 1. From (2.4) it follows that (3.50)

lim HB(xn, e) = a lim HB(ck, e) + (1 - a) lim 7 7 s e ) n—>» k~>+oo —»

(cf. the argument following (3.9) in [13]). We now construct other sequences {x'n} for which the corresponding a' takes any value in [0, 1]. This is done as follows. By (2.13), (3.51)

aif(crTxn) =

aN+r(xn),

and for fixed N, n, (3.52)

lim ajf+r{xn) = 0,

lim aN+r{xn)

r—• oo

r—• — co

= 1.

Moreover, by (2.13), (3.49), and (3.44), (3.53)

0 < aN+r{xn)

- aN+r+1(xn)

= H§{c-N-'xn,

= H§{xn,