Branching Processes
Branching processes form one of the classical fields of applied probability and are still an active area of research. The field has by now grown so large and diverse that a complete and unified treat ment is hardly possible anymore, let alone in one volu
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Springer Science+Business Media, LLC
S. Asmussen H. Hering
Branching Processes
1983 Springer Science+Business Media, LLC
Authors: Sj!lren Asmussen Institute of Mathematical Stochastik University of Copenhagen 5, Universitetsparken 2100 Copenhagen ~. Denmark Heinrich Hering Institut ffir Mathematische Statistik Universit~t Gtlttingen Lotzestr. 13 3400 Gtlttingen, West Germany CIP-Kurztitelaufnahme der Deutschen Bibliothek Asmussen, Sj!lren: Branchina nrocesses 1 S. Asmussen ; H. Hering. - Boston ; Basel ; Stuttgart ; Birkhauser, 1983. (Progress in probability and statistics ; Vol. 3)
ISBN 978-1-4615-8155-0 (eBook) ISBN 978-0-8176-3122-2 DOI 10.1007/978-1-4615-8155-0
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Hering, Heinrich: ; GT
Library of Congress Cataloging in Publication Data Asmussen, Sl!lren. Branching processes. (Progress in probability and statistics ; V. 3) Bibliography: 1. Branching processes. I. Hering, H. (Heinrich), . II. Title. III. Seri es. 194082-22704 519.2'34 QA274.76.A78 1983 ISBN 978-0-8176-3122-2
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission of the copyright owner. © Springer Science+Business Media New York 1983 Originally published by Birkhäuser Boston in 1983 Softcover reprint of the hardcover 1st edition 1983
ISBN 978-0-8176-3122-2
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TABLE OF CONTENTS PART A: INTRODUCTION Chapter I: Branching phenomena and models 1. Simple branching processes ................................ 2 2. p-type processes 7 Age dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3. 4. General processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Bibliographical notes ..................................... 16 PART B: SIMPLE BRANCHING PROCESSES Chapter II: The Galton-Watson process: Probabilistic methods 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The Kesten-Stigum theorem ................................. 3. Finer limit theorems: Finite offspring variance .......... 4. Finer limit theorems: Infinite offspring variance ........ 5. The Seneta-Heyde theorem .................................. 6. Immigration •.............................................. Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18 23 28 36 43 50 54
Chapter III: The Galton-Watson process: Analytic methods 1. Subcritical processes: Yaglom's theorem ..............•... 56 2. Arbitrary initial distributions and invariant measures ... 65 3. Critical processes: The exponential limit theorem ........ 74 4. Local limit theorems for critica1 processes ...........•.. 78 5. Supercritical processes: Basic convergence result ........ 83 6. Further properties of the limiting distribution .......... 89 7. Local limit theorem for supercritical processes .......... 97 8. Immigration ........................................
