Controlled Branching Processes

The purpose of this book is to provide a comprehensive discussion of the available results for discrete time branching processes with random control functions. The independence of individuals’ reproduction is a fundamental assumption in the classical branching processes. Alternatively, the controlled branching processes (CBPs) allow the number of reproductive individuals in one generation to decrease or increase depending on the size of the previous generation.

Generating a wide range of behaviors, the CBPs have been successfully used as modeling tools in diverse areas of applications.

Foreword ix

Preface xi

Chapter 1 Classical Branching Models 1

1.1 Bienaymé–Galton–Watson process 1

1.1.1 Moments and probability of extinction 4

1.1.2 Limit theorems 9

1.2 Processes with unrestricted immigration 17

1.2.1 Limit theorems 21

1.2.2 Critical process with decreasing to zero immigration 25

1.3 Processes with immigration after empty generation only 29

1.3.1 Limit theorems 31

1.3.2 Critical process with decreasing to zero immigration 36

1.4 Background and bibliographical notes 40

Chapter 2 Branching Processes with Migration 43

2.1 Galton–Watson process with migration 43

2.2 Limit theorems 47

2.2.1 Non-critical processes 47

2.2.2 Critical processes with non-negative migration mean 49

2.2.3 Critical processes with negative migration mean 52

2.3 Regeneration and migration 55

2.3.1 Alternating regenerative processes 56

2.3.2 An extension of Galton–Watson processes with migration 58

2.4 Background and bibliographical notes 62

Chapter 3 CB Processes: Extinction 65

3.1 Definition of processes and basic properties 65

3.1.1 Basic properties 69

3.1.2 Probability generating functions and moments 73

3.2 Extinction probability 75

3.2.1 Subcritical processes 76

3.2.2 Supercritical processes 78

3.2.3 Critical processes 84

3.3 Background and bibliographical notes 91

Chapter 4 CB Processes: Limit Theorems 95

4.1 Subcritical processes 95

4.2 Critical processes 100

4.2.1 Extinction is not certain 101

4.2.2 Extinction is certain 109

4.2.3 Feller diffusion approximation 110

4.3 Supercritical processes 115

4.3.1 Almost sure convergence 117

4.3.2 L¹–convergence 118

4.3.3 L²–convergence 121

4.4 Background and bibliographical notes 125

Chapter 5 Statistics of CB Processes 127

5.1 Maximum likelihood estimation 127

5.1.1 MLE based on entire family tree up to nth generation 130

5.1.2 EM algorithms for incomplete data 146

5.1.3 Simulated example 152

5.2 Conditional weighted least squares estimation 158

5.2.1 Subcritical processes 159

5.2.2 Critical processes 161

5.2.3 Supercritical processes 166

5.3 Minimum disparity estimation 169

5.4 Bayesian inference 171

5.4.1 Estimation based on entire family tree up to nth generation 172

5.4.2 MCMC algorithms for incomplete data 173

5.5 Background and bibliographical notes 176

Appendices 179

Appendix 1 181

Appendix 2 185

Appendix 3 191

Appendix 4 195

Bibliography 197

Index 209

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