Branching Processes

300 Pages·1972·7.427 MB·English

by  Krishna B. Athreya, Peter E. Ney (auth.)

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 IIe rausgeber
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, re
production 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, re
production 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 pro
cess, etc. We have tried to avoid the rather uneconomical and un
enlightening 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 age
dependent) 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 pro
cesses. 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 biblio
graphy.
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 off~(s) in the Noncritical Cases 38
Part D. Further Ramifications 47
12. Decomposition of the Supercritical Branching Process. 47
13. Second Order Properties of Z" 1m 53
II • • • •
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