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Discrete Time Branching Processes in Random Environment

Book Description

Branching processes are stochastic processes which represent the reproduction of particles, such as individuals within a population, and thereby model demographic stochasticity. In branching processes in random environment (BPREs), additional environmental stochasticity is incorporated, meaning that the conditions of reproduction may vary in a random fashion from one generation to the next.

This book offers an introduction to the basics of BPREs and then presents the cases of critical and subcritical processes in detail, the latter dividing into weakly, intermediate, and strongly subcritical regimes.

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Preface
  5. List of Notations
  6. 1 Branching Processes in Varying Environment
    1. 1.1. Introduction
    2. 1.2. Extinction probabilities
    3. 1.3. Almost sure convergence
    4. 1.4. Family trees
    5. 1.5. Notes
  7. 2 Branching Processes in Random Environment
    1. 2.1. Introduction
    2. 2.2. Extinction probabilities
    3. 2.3. Exponential growth in the supercritical case
    4. 2.4. Three subcritical regimes
    5. 2.5. The strictly critical case
    6. 2.6. Notes
  8. 3 Large Deviations for BPREs
    1. 3.1. Introduction
    2. 3.2. A tail estimate for branching processes in a varying environment
    3. 3.3. Proof of Theorem 3.1
    4. 3.4. Notes
  9. 4 Properties of Random Walks
    1. 4.1. Introduction
    2. 4.2. Sparre-Andersen identities
    3. 4.3. Spitzer identity
    4. 4.4. Applications of Sparre-Andersen and Spitzer identities
    5. 4.5. Notes
  10. 5 Critical BPREs: the Annealed Approach
    1. 5.1. Introduction
    2. 5.2. Changes of measures
    3. 5.3. Properties of the prospective minima
    4. 5.4. Survival probability
    5. 5.5. Limit theorems for the critical case (annealed approach)
    6. 5.6. Environment providing survival
    7. 5.7. Convergence of log Zn
    8. 5.8. Notes
  11. 6 Critical BPREs: the Quenched Approach
    1. 6.1. Introduction
    2. 6.2. Changes of measures
    3. 6.3. Probability of survival
    4. 6.4. Yaglom limit theorems
    5. 6.5. Discrete limit distributions
    6. 6.6. Notes
  12. 7 Weakly Subcritical BPREs
    1. 7.1. Introduction
    2. 7.2. The probability measures and
    3. 7.3. Proof of theorems
    4. 7.4. Notes
  13. 8 Intermediate Subcritical BPREs
    1. 8.1. Introduction
    2. 8.2. Proof of Theorems 8.1 to 8.3
    3. 8.3. Further limit results
    4. 8.4. Conditioned family trees
    5. 8.5. Proof of Theorem 8.4
    6. 8.6. Notes
  14. 9 Strongly Subcritical BPREs
    1. 9.1. Introduction
    2. 9.2. Survival probability and Yaglom-type limit theorems
    3. 9.3. Environments providing survival and dynamics of the population size
    4. 9.4. Notes
  15. 10 Multi-type BPREs
    1. 10.1. Introduction
    2. 10.2. Supercritical MBPREs
    3. 10.3. The survival probability of subcritical and critical MBPREs
    4. 10.4. Functional limit theorem in the critical case
    5. 10.5. Subcritical multi-type case
  16. Appendix
    1. A.1. Examples of slowly varying functions
  17. Bibliography
  18. Index
  19. End User License Agreement