Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Beriicksichtigung der Anwendungsgebiete Band 196 llerausgegeben von J. L. Doob. A. Grothendieck. E. Heinz F. Hirzebruch E. Hopf. W Maak. S. Mac Lane W Magnus J. K. Moser M. M. Postnikov. F. K. Schmidt D. S. Scott K. Stein Geschaftsfiihrende II erausgeber B. Eckmann und B. L. van der Waerden
K. B. Athreya. P. E. Ney Branching Processes Springer-Verlag Berlin Heidelberg New York 1972
Krishna B. Athreya Associate Professor of Mathematics University of Wisconsin, Madison, Wisconsin Peter E. Ney Professor of Mathematics University of Wisconsin, Madison, Wisconsin Geschaftsfiihrende Herausgeber: B. Eckmann Eidgeniissische Technische Hochschule Ziirich B. L. van der Waerden Mathematisches Institut der Universitiit Ziirich AMS Subject Classifications (1970) Primary 60J80, 60K99, 60F99 Secondary 60J85, 39A20, 45M05, 60J 45, 92A 15 ISBN -13: 978-3-642-65373-5 e-isbn -13: 978-3-642-65371-1 DOl: 10.1007/978-3-642-65371-1 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data qanks. Under 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. by Springer Verlag Berlin Heidelberg 1972. Library of Congress Catalog Card Number 72-75819. Softcover reprint of the hardcover 1st edition 1972
K. B. Athreya. P. E. Ney Branching Processes Springer-Verlag New York Heidelberg Berlin 1972
Krishna B. Athreya Associate Professor of Mathematics University of Wisconsin, Madison, Wisconsin Peter E. Ney Professor of Mathematics University of Wisconsin, Madison, Wisconsin Geschliftsfiihrende Herausgeber: B. Eckmann Eidgeniissische Technische Hochschule Ziirich B. L. van der Waerden Mathematisches Institut der Universit.t Ziirich AMS Subject Classifications (1970) Primary 60 J 80, 60 K 99, 60 F 99 Secondary 60J 85, 39A20, 45 M05, 60J 45, 92A 15 ISBN -13: 978-3-642-65373-5 e-isbn -13: 978-3-642-65371-1 DOl: 10.1007/978-3-642-65371-1 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data hanks. Under 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. by Springer Verlag Berlin Heidelberg 1972. Library of Congress Catalog Card Number 72-75819. Softcover reprint of the hardcover 1st edition 1972
Dedicated to our parents
Preface The purpose of this book is to give a unified treatment of the limit theory of branching processes. Since the publication of the important book of T E. Harris (Theory of Branching Processes, Springer, 1963) the subject has developed and matured significantly. Many of the classical limit laws are now known in their sharpest form, and there are new proofs that give insight into the results. Our work deals primarily with this decade, and thus has very little overlap with that of Harris. Only enough material is repeated to make the treatment essentially self-contained. For example, certain foundational questions on the construction of processes, to which we have nothing new to add, are not developed. There is a natural classification of branching processes according to their criticality condition, their time parameter, the single or multi-type particle cases, the Markovian or non-markovian character of the process, etc. We have tried to avoid the rather uneconomical and unenlightening approach of treating these categories independently, and by a series of similar but increasingly complicated techniques. The basic Galton-Watson process is developed in great detail in Chapters I and II. In the subsequent treatment of the continuous time (Markov and agedependent) cases in Chapters III and IV, we try wherever possible to reduce analogous questions to their Galton-Watson counterparts; and then concentrate on the genuinely new or different aspects of these processes. We hope that this gives the subject a more unified aspect. In our development we give a number of new proofs of known results; and also some new results which appear here for the first time. Although we have included a chapter on applications and special processes (Chapter VI), these are a reflection of our own interests, and there is no attempt to catalogue or even dent the great variety of special models in physics and biology that have been investigated in recent years. Rather, our emphasis is on the basic techniques, and it is our hope that this volume will bring the reader to the point where he can pursue the literature and his own research in the subject. In this connection we have included sections on complements and problems, which may suggest new work.
VIII Preface With the exception of Chapter V which deals with a finite number of distinct particle types, we have concentrated on single type processes. There is an extensive literature on branching processes on general state spaces which is outside the spirit and scope of this book. Readers interested in this subject are referred to the papers of Ikeda, Nagasawa, Watanabe, Jirina, Moyal, Mullikin, and Skorohod listed in the bibliography. The prerequisites for reading this book are analysis and probability at about the first year graduate level. The reader should, for example, have some familiarity with Markov chains at the level of K. L. Chung's book (Springer, 1967); the martingale theorem; renewal theory and analytical probability at the level of W. Feller, Vol. II (Wiley, 1966). Some of the more technical sections can be skipped over without impairing continuity. A possible sequence to follow at a first reading is Chapter I Chapter III, sections 1-8, Chapter IV Chapter V, omitting section 8, Chapter VI, selections according to taste. Acknowledgements We want to express our appreciation to our collegues and teachers J. Chover, T. Harris, A. Joffe, S. Karlin, H. Kesten, and F. Spitzer for their help and influence on our work. Their contributions to the theory of branching processes permeate these pages. We also wish to extend our sincere thanks to J. L. Doob, as editor of this series, for his counsel and valuable sl.lggestions. Many others have given us assistance. J. Lamperti and J. Williamson read parts of the manuscript and made helpful comments. Among our former and present students, M. Goldstein prepared the original seminar notes from which the book started; J. Foster, K. B. Erickson, W. Esty, and R. Alexander corrected vast numbers of errors; W. Esty prepared the bibliography and index and helped extensively in the preparation of the final manuscript. Judy Brickner did a superb job of typing the manuscript, and then cheerfully saw it through revisions and re-revisions. Finally, we would like to thank our wives for their moral support throughout this long project. Madison, Wisconsin * September, 1971 Krishna B. Athreya. Peter E. Ney * K. Athreya is now at the Indian Institute of Science in Bangalore, India.
Table of Contents Chapter I. The Galton-Watson Process Part A. Preliminaries.. 1. The Basic Setting. 1 2. Moments.... 4 3. Elementary Properties of Generating Functions. 4 4. An Important Example..... 6 5. Extinctlon Probability..... 7 Part B. A First Look at Limit Theorems 8 6. Motivating Remarks.. 8 7. Ratio Theorems.... 11 8. Conditioned Limit Laws 15 9. The Exponential Limit Law for the Critical Process 19 Part C. Finer Limit Theorems............ 24 10. Strong Convergence in the Supercritical Case... 24 11. Geometric Convergence o f f ~ in ( the s ) Noncritical Cases 38 Part D. Further Ramifications 47 12. Decomposition of the Supercritical Branching Process. 47 13. Second Order Properties of Z" 1m II 53 14. The Q-Process............. 56 15. More on Conditioning; Limiting Diffusions 61 Complements and Problems I. 62 Chapter II. Potential Theory 1. Introduction.............. 66 2. Stationary Measures: Existence, Uniqueness, and Representation............ 67 3. The Local Limit Theorem for the Critical Case.. 73 4. The Local Limit Theorem for the Supercritical Case 79
x Table of Contents 5. Further Properties of W; A Sharp Global Limit Law; Positivity of the Density.......... 82 6. Asymptotic Properties of Stationary Measures 87 7. Green Function Behavior. 90 8. Harmonic Functions... 93 9. The Space-Time Boundary 98 Complements and Problems II.. 99 Chapter III. One Dimensional Continuous Time Markov Branching Processes 1. Definition...... 102 2. Construction.......... 103 3. Generating Functions....... 106 4. Extinction Probability and Moments 107 5. Examples: Binary Fission, Birth and Death Process. 109 6. The Embedded Galton-Watson Process and Applications to Moments......... 110 7. Limit Theorems....... 111 8. More on Generating Functions. 115 9. Split Times......... 118 10. Second Order Properties... 123 11. Constructions Related to Poisson Processes 125 12. The Embeddability Problem. 130 Complements and Problems III........ 136 Chapter IV. Age-Dependent Processes 1. Introduction................ 137 2. Existence and Uniqueness.......... 139 3. Comparison with Galton-Watson Process; Embedded Generation Process; Extinction Probability 141 4. Renewal Theory............... 144 5. Moments.................. 150 6. Asymptotic Behavior of F(s, t) in the Critical Case 158 7. Asymptotic Behavior of F(s, t) when m=l= 1: The Malthusian Case............... 162 8. Asymptotic Behavior of F(s, t) when m =/=1: Sub-Exponential Case...... :................ 168 9. The Exponential Limit Law in the Critical Case..... 169 10. The Limit Law for the Subcritical Age-Dependent Process 170 11. Limit Theorems for the Supercritical Case. 171 Complements and Problems IV............... 177
Table of Contents XI Chapter V. Multi-Type Branching Processes 1. Introduction and Definitions..... 181 2. Moments and the Frobenius Theorem. 184 3. Extinction Probability and Transience. 186 4. Limit Theorems for the Subcritical Case. 186 5. Limit Theorems for the Critical Case.. 189 6. The Supercritical Case and Geometric Growth 192 7. The Continuous Time, Multitype Markov Case. 199 8. Linear Functionals of Supercritical Processes.. 209 9. Embedding of Urn Schemes into Continuous Time Markov Branching Processes........ 219 10. The Multitype Age-Dependent Process 225 Complements and Problems V....... 227 Chapter VI. Special Processes 1. A One Dimensional Branching Random Walk 2. Cascades; Distributions of Generations 3. Branching Diffusions........... 4. Martingale Methods........... 5. Branching Processes with Random Environments. 6. Continuous State Branching Processes. 7. Immigration...... 8. Instability....... Complements and Problems VI Bibliography. List of Symbols Author Index Subject Index 230 239 242 246 249 257 262 266 267 268 282 283 285