Miguel González Velasco is Associate Professor at the University of Extremadura in Spain. His research interests lie in the theory of branching processes and its application to genetics, epidemiology and population dynamics.Inés M. del Puerto García is Associate Professor at the University of Extremadura in Spain. Her research interests include the study of the probabilistic and inferential theory on branching processes.George P. Yanev is Associate Professor at the University of Texas Rio Grande Valley in the USA. His research interests include branching processes and characterizations of probability distributions.

• Foreword ixPreface xiChapter 1 Classical Branching Models 11.1 Bienaymé–Galton–Watson process 11.1.1 Moments and probability of extinction 41.1.2 Limit theorems 91.2 Processes with unrestricted immigration 171.2.1 Limit theorems 211.2.2 Critical process with decreasing to zero immigration 251.3 Processes with immigration after empty generation only 291.3.1 Limit theorems 311.3.2 Critical process with decreasing to zero immigration 361.4 Background and bibliographical notes 40Chapter 2 Branching Processes with Migration 432.1 Galton–Watson process with migration 432.2 Limit theorems 472.2.1 Non-critical processes 472.2.2 Critical processes with non-negative migration mean 492.2.3 Critical processes with negative migration mean 522.3 Regeneration and migration 552.3.1 Alternating regenerative processes 562.3.2 An extension of Galton–Watson processes with migration 582.4 Background and bibliographical notes 62Chapter 3 CB Processes: Extinction 653.1 Definition of processes and basic properties 653.1.1 Basic properties 693.1.2 Probability generating functions and moments 733.2 Extinction probability 753.2.1 Subcritical processes 763.2.2 Supercritical processes 783.2.3 Critical processes 843.3 Background and bibliographical notes 91Chapter 4 CB Processes: Limit Theorems 954.1 Subcritical processes 954.2 Critical processes 1004.2.1 Extinction is not certain 1014.2.2 Extinction is certain 1094.2.3 Feller diffusion approximation 1104.3 Supercritical processes 1154.3.1 Almost sure convergence 1174.3.2 L1–convergence 1184.3.3 L2–convergence 1214.4 Background and bibliographical notes 125Chapter 5 Statistics of CB Processes 1275.1 Maximum likelihood estimation 1275.1.1 MLE based on entire family tree up to nth generation 1305.1.2 EM algorithms for incomplete data 1465.1.3 Simulated example 1525.2 Conditional weighted least squares estimation 1585.2.1 Subcritical processes 1595.2.2 Critical processes 1615.2.3 Supercritical processes 1665.3 Minimum disparity estimation 1695.4 Bayesian inference 1715.4.1 Estimation based on entire family tree up to nth generation 1725.4.2 MCMC algorithms for incomplete data 1735.5 Background and bibliographical notes 176Appendices 179Appendix 1 181Appendix 2 185Appendix 3 191Appendix 4 195Bibliography 197Index 209

Miguel González Velasco is Associate Professor at the University of Extremadura in Spain. His research interests lie in the theory of branching processes and its application to genetics, epidemiology and population dynamics.Inés M. del Puerto García is Associate Professor at the University of Extremadura in Spain. Her research interests include the study of the probabilistic and inferential theory on branching processes.George P. Yanev is Associate Professor at the University of Texas Rio Grande Valley in the USA. His research interests include branching processes and characterizations of probability distributions.

• Foreword ixPreface xiChapter 1 Classical Branching Models 11.1 Bienaymé–Galton–Watson process 11.1.1 Moments and probability of extinction 41.1.2 Limit theorems 91.2 Processes with unrestricted immigration 171.2.1 Limit theorems 211.2.2 Critical process with decreasing to zero immigration 251.3 Processes with immigration after empty generation only 291.3.1 Limit theorems 311.3.2 Critical process with decreasing to zero immigration 361.4 Background and bibliographical notes 40Chapter 2 Branching Processes with Migration 432.1 Galton–Watson process with migration 432.2 Limit theorems 472.2.1 Non-critical processes 472.2.2 Critical processes with non-negative migration mean 492.2.3 Critical processes with negative migration mean 522.3 Regeneration and migration 552.3.1 Alternating regenerative processes 562.3.2 An extension of Galton–Watson processes with migration 582.4 Background and bibliographical notes 62Chapter 3 CB Processes: Extinction 653.1 Definition of processes and basic properties 653.1.1 Basic properties 693.1.2 Probability generating functions and moments 733.2 Extinction probability 753.2.1 Subcritical processes 763.2.2 Supercritical processes 783.2.3 Critical processes 843.3 Background and bibliographical notes 91Chapter 4 CB Processes: Limit Theorems 954.1 Subcritical processes 954.2 Critical processes 1004.2.1 Extinction is not certain 1014.2.2 Extinction is certain 1094.2.3 Feller diffusion approximation 1104.3 Supercritical processes 1154.3.1 Almost sure convergence 1174.3.2 L1–convergence 1184.3.3 L2–convergence 1214.4 Background and bibliographical notes 125Chapter 5 Statistics of CB Processes 1275.1 Maximum likelihood estimation 1275.1.1 MLE based on entire family tree up to nth generation 1305.1.2 EM algorithms for incomplete data 1465.1.3 Simulated example 1525.2 Conditional weighted least squares estimation 1585.2.1 Subcritical processes 1595.2.2 Critical processes 1615.2.3 Supercritical processes 1665.3 Minimum disparity estimation 1695.4 Bayesian inference 1715.4.1 Estimation based on entire family tree up to nth generation 1725.4.2 MCMC algorithms for incomplete data 1735.5 Background and bibliographical notes 176Appendices 179Appendix 1 181Appendix 2 185Appendix 3 191Appendix 4 195Bibliography 197Index 209

Miguel González Velasco is Associate Professor at the University of Extremadura in Spain. His research interests lie in the theory of branching processes and its application to genetics, epidemiology and population dynamics.Inés M. del Puerto García is Associate Professor at the University of Extremadura in Spain. Her research interests include the study of the probabilistic and inferential theory on branching processes.George P. Yanev is Associate Professor at the University of Texas Rio Grande Valley in the USA. His research interests include branching processes and characterizations of probability distributions.

• Foreword ixPreface xiChapter 1 Classical Branching Models 11.1 Bienaymé–Galton–Watson process 11.1.1 Moments and probability of extinction 41.1.2 Limit theorems 91.2 Processes with unrestricted immigration 171.2.1 Limit theorems 211.2.2 Critical process with decreasing to zero immigration 251.3 Processes with immigration after empty generation only 291.3.1 Limit theorems 311.3.2 Critical process with decreasing to zero immigration 361.4 Background and bibliographical notes 40Chapter 2 Branching Processes with Migration 432.1 Galton–Watson process with migration 432.2 Limit theorems 472.2.1 Non-critical processes 472.2.2 Critical processes with non-negative migration mean 492.2.3 Critical processes with negative migration mean 522.3 Regeneration and migration 552.3.1 Alternating regenerative processes 562.3.2 An extension of Galton–Watson processes with migration 582.4 Background and bibliographical notes 62Chapter 3 CB Processes: Extinction 653.1 Definition of processes and basic properties 653.1.1 Basic properties 693.1.2 Probability generating functions and moments 733.2 Extinction probability 753.2.1 Subcritical processes 763.2.2 Supercritical processes 783.2.3 Critical processes 843.3 Background and bibliographical notes 91Chapter 4 CB Processes: Limit Theorems 954.1 Subcritical processes 954.2 Critical processes 1004.2.1 Extinction is not certain 1014.2.2 Extinction is certain 1094.2.3 Feller diffusion approximation 1104.3 Supercritical processes 1154.3.1 Almost sure convergence 1174.3.2 L1–convergence 1184.3.3 L2–convergence 1214.4 Background and bibliographical notes 125Chapter 5 Statistics of CB Processes 1275.1 Maximum likelihood estimation 1275.1.1 MLE based on entire family tree up to nth generation 1305.1.2 EM algorithms for incomplete data 1465.1.3 Simulated example 1525.2 Conditional weighted least squares estimation 1585.2.1 Subcritical processes 1595.2.2 Critical processes 1615.2.3 Supercritical processes 1665.3 Minimum disparity estimation 1695.4 Bayesian inference 1715.4.1 Estimation based on entire family tree up to nth generation 1725.4.2 MCMC algorithms for incomplete data 1735.5 Background and bibliographical notes 176Appendices 179Appendix 1 181Appendix 2 185Appendix 3 191Appendix 4 195Bibliography 197Index 209