Controlled Branching Processes

237 Pages·2018·3.621 MB·English

by  Del Puerto Garcia, Ines Maria, Velasco, Miguel Gonzalez, Yanev, George Petrov

Controlled Branching Processes
To Pilar and our daughter, Pilar,
and sons, Miguel and Manuel

To Raúl and our sons, Jorge and Marcos,
and daughter Sara

To Reneta and Petar
Branching Processes, Branching Random Walks and
Branching Particle Fields Set
coordinated by
Elena Yarovaya
Volume 2
Controlled Branching
Processes

Miguel González Velasco
Inés M. del Puerto García
George P. Yanev
First published 2018 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as
permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced,
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undermentioned address:
ISTE Ltd John Wiley & Sons, Inc.
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London SW19 4EU Hoboken, NJ 07030
UK USA
www.iste.co.uk www.wiley.com

© ISTE Ltd 2018
The rights of Miguel González Velasco, Inés M. del Puerto García and George P. Yanev to be identified
as the authors of this work have been asserted by them in accordance with the Copyright, Designs and
Patents Act 1988.
Library of Congress Control Number: 2017956887

British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-78630-253-3
Contents
Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Chapter 1. Classical Branching Models . . . . . . . . . . . . . . . 1
1.1.Bienaymé–Galton–Watsonprocess . . . . . . . . . . . . . . . . 1
1.1.1.Momentsandprobabilityofextinction . . . . . . . . . . . . 4
1.1.2.Limittheorems . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.Processeswithunrestrictedimmigration . . . . . . . . . . . . . 17
1.2.1.Limittheorems . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.2.2.Criticalprocesswithdecreasingtozeroimmigration . . . . 25
1.3.Processeswithimmigrationafteremptygenerationonly . . . . 29
1.3.1.Limittheorems . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.3.2.Criticalprocesswithdecreasingtozeroimmigration . . . . 36
1.4.Backgroundandbibliographicalnotes . . . . . . . . . . . . . . . 40
Chapter 2. Branching Processes with Migration . . . . . . . . . 43
2.1.Galton–Watsonprocesswithmigration . . . . . . . . . . . . . . 43
2.2.Limittheorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.2.1.Non-criticalprocesses . . . . . . . . . . . . . . . . . . . . . . 47
2.2.2.Criticalprocesseswithnon-negativemigrationmean . . . . 49
2.2.3.Criticalprocesseswithnegativemigrationmean . . . . . . . 52
2.3.Regenerationandmigration. . . . . . . . . . . . . . . . . . . . . 55
2.3.1.Alternatingregenerativeprocesses . . . . . . . . . . . . . . . 56
vi ControlledBranchingProcesses
2.3.2.AnextensionofGalton–Watsonprocesses
withmigration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.4.Backgroundandbibliographicalnotes . . . . . . . . . . . . . . . 62
Chapter 3. CB Processes: Extinction . . . . . . . . . . . . . . . . 65
3.1.Definitionofprocessesandbasicproperties . . . . . . . . . . . 65
3.1.1.Basicproperties . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.1.2.Probabilitygeneratingfunctionsandmoments . . . . . . . . 73
3.2.Extinctionprobability . . . . . . . . . . . . . . . . . . . . . . . . 75
3.2.1.Subcriticalprocesses . . . . . . . . . . . . . . . . . . . . . . 76
3.2.2.Supercriticalprocesses . . . . . . . . . . . . . . . . . . . . . 78
3.2.3.Criticalprocesses . . . . . . . . . . . . . . . . . . . . . . . . 84
3.3.Backgroundandbibliographicalnotes . . . . . . . . . . . . . . . 91
Chapter 4. CB Processes: Limit Theorems . . . . . . . . . . . . 95
4.1.Subcriticalprocesses . . . . . . . . . . . . . . . . . . . . . . . . 95
4.2.Criticalprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.2.1.Extinctionisnotcertain . . . . . . . . . . . . . . . . . . . . . 101
4.2.2.Extinctioniscertain . . . . . . . . . . . . . . . . . . . . . . . 109
4.2.3.Fellerdiffusionapproximation . . . . . . . . . . . . . . . . . 110
4.3.Supercriticalprocesses . . . . . . . . . . . . . . . . . . . . . . . 115
4.3.1.Almostsureconvergence . . . . . . . . . . . . . . . . . . . . 117
1
4.3.2.L –convergence . . . . . . . . . . . . . . . . . . . . . . . . . 118
2
4.3.3.L –convergence . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.4.Backgroundandbibliographicalnotes . . . . . . . . . . . . . . . 125
Chapter 5. Statistics of CB Processes . . . . . . . . . . . . . . . . 127
5.1.Maximumlikelihoodestimation . . . . . . . . . . . . . . . . . . 127
5.1.1.MLEbasedonentirefamilytreeuptonthgeneration . . . . 130
5.1.2.EMalgorithmsforincompletedata . . . . . . . . . . . . . . 146
5.1.3.Simulatedexample . . . . . . . . . . . . . . . . . . . . . . . 152
5.2.Conditionalweightedleastsquaresestimation . . . . . . . . . . 158
5.2.1.Subcriticalprocesses . . . . . . . . . . . . . . . . . . . . . . 159
5.2.2.Criticalprocesses . . . . . . . . . . . . . . . . . . . . . . . . 161
5.2.3.Supercriticalprocesses . . . . . . . . . . . . . . . . . . . . . 166
5.3.Minimumdisparityestimation . . . . . . . . . . . . . . . . . . . 169
Contents vii
5.4.Bayesianinference. . . . . . . . . . . . . . . . . . . . . . . . . . 171
5.4.1.Estimationbasedonentirefamilytreeuptonthgeneration . 172
5.4.2.MCMCalgorithmsforincompletedata . . . . . . . . . . . . 173
5.5.Backgroundandbibliographicalnotes . . . . . . . . . . . . . . . 176
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Appendix 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Appendix 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
Appendix 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209