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Discrete time branching processes in random environment

Author: Götz Kersting; V A Vatutin
Publisher: London : ISTE Ltd ; Hoboken, NJ : John Wiley & Sons, Inc., 2017.
Series: Mathematics and statistics series (ISTE)., Branching processes, branching random walks and branching particle fields set ;, v. 1.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
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  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Kersting, Götz.
Discrete Time Branching Processes in Random Environment.
London : ISTE Ltd ; Hoboken, NJ : John Wiley & Sons, Inc., 2017
(OCoLC)1004760993
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Götz Kersting; V A Vatutin
ISBN: 9781119473558 1119473551 9781119452898 1119452899 9781119473770 1119473772
OCLC Number: 1009042545
Description: 1 online resource
Contents: ""Cover""; ""Half-Title Page""; ""Title Page""; ""Copyright Page""; ""Contents""; ""Preface""; ""List of Notations""; ""1. Branching Processes in Varying Environment""; ""1.1. Introduction""; ""1.2. Extinction probabilities""; ""1.3. Almost sure convergence""; ""1.4. Family trees""; ""1.4.1. Construction of the Geiger tree""; ""1.4.2. Construction of the size-biased tree T*""; ""1.5. Notes""; ""2. Branching Processes in Random Environment""; ""2.1. Introduction""; ""2.2. Extinction probabilities""; ""2.3. Exponential growth in the supercritical case""; ""2.4. Three subcritical regimes"" ""2.5. The strictly critical case""""2.6. Notes""; ""3. Large Deviations for BPREs""; ""3.1. Introduction""; ""3.2. A tail estimate for branching processes in a varying environment""; ""3.3. Proof of Theorem 3.1""; ""3.4. Notes""; ""4. Properties of Random Walks""; ""4.1. Introduction""; ""4.2. Sparre-Andersen identities""; ""4.3. Spitzer identity""; ""4.4. Applications of Sparre-Andersen and Spitzer identities""; ""4.4.1. Properties of ladder epochs and ladder heights""; ""4.4.2. Tail distributions of ladder epochs""; ""4.4.3. Some renewal functions"" ""4.4.4. Asymptotic properties of Ln and Mn""""4.4.5. Arcsine law""; ""4.4.6. Large deviations for random walks""; ""4.5. Notes""; ""5. Critical BPREs: the Annealed Approach""; ""5.1. Introduction""; ""5.2. Changes of measures""; ""5.3. Properties of the prospective minima""; ""5.4. Survival probability""; ""5.5. Limit theorems for the critical case (annealed approach)""; ""5.6. Environment providing survival""; ""5.7. Convergence of log Zn""; ""5.8. Notes""; ""6. Critical BPREs: the Quenched Approach""; ""6.1. Introduction""; ""6.2. Changes of measures""; ""6.3. Probability of survival"" ""6.4. Yaglom limit theorems""""6.4.1. The population size at non-random moments""; ""6.4.2. The population size at moments nt, 0 nt""; ""6.5. Discrete limit distributions""; ""6.6. Notes""; ""7. Weakly Subcritical BPREs""; ""7.1. Introduction""; ""7.2. The probability measures P+ and Pâ#x88;#x92;""; ""7.3. Proof of theorems""; ""7.3.1. Proof of Theorem 7.1""; ""7.3.2. Proof of Theorem 7.2""; ""7.3.3. Proof of Theorem 7.3""; ""7.4. Notes""; ""8. Intermediate Subcritical BPREs"" ""8.1. Introduction""""8.2. Proof of Theorems 8.1 to 8.3""; ""8.3. Further limit results""; ""8.4. Conditioned family trees""; ""8.5. Proof of Theorem 8.4""; ""8.6. Notes""; ""9. Strongly Subcritical BPREs""; ""9.1. Introduction""; ""9.2. Survival probability and Yaglom-type limit theorems""; ""9.3. Environments providing survival and dynamics of the population size""; ""9.3.1. Properties of the transition matrix P*""; ""9.3.2. Proof of Theorem 9.2""; ""9.3.3. Proof of Theorem 9.3""; ""9.4. Notes""; ""10. Multi-type BPREs""; ""10.1. Introduction""; ""10.2. Supercritical MBPREs""
Series Title: Mathematics and statistics series (ISTE)., Branching processes, branching random walks and branching particle fields set ;, v. 1.
Responsibility: Götz Kersting, Vladimir Vatutin.

Abstract:

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.

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