Discrete Time Branching Processes in Random Environment

✍ Scribed by Kersting, Götz; Vatutin, Vladimir

Publisher: ISTE Ltd ; Hoboken
Year: 2017
Tongue: English
Leaves: 295
Series: Mathematics and statistics series (ISTE) Vol. 1
Category: Library

No coin nor oath required. For personal study only.

✦ Synopsis

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...

Abstract:

Branching processes are stochastic processes which represent the reproduction of particles, such as individuals within a population, and thereby model demographic stochasticity. �Read more...

✦ Table of Contents

Content: ""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""

✦ Subjects

Branching processes.;MATHEMATICS / Applied.;MATHEMATICS / Probability & Statistics / General.

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