Discrete Time Branching Processes in Random Environment 1st Edition by Götz Kersting, Vladimir Vatutin ISBN 978-1786302526 1786302527
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Product details:
ISBN 10: 1786302527
ISBN 13: 978-1786302526
Author: Götz Kersting, Vladimir Vatutin
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:
Chapter 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
Chapter 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
Chapter 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
Chapter 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 L₁₂ and Mn
4.4.5. Arcsine law
4.4.6. Large deviations for random walks
4.5. Notes
Chapter 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
Chapter 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 < t < 1
6.4.3. The number of particles at moment (n) ≤ nt
6.4.4. The number of particles at moment (n) > nt
6.5. Discrete limit distributions
6.6. Notes
Chapter 7. Weakly Subcritical BPRES
7.1. Introduction
7.2. The probability measures Pt and P
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
Chapter 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
Chapter 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
Chapter 10. Multi-type BPRES
10.1. Introduction
10.2. Supercritical MBPRES
10.3. The survival probability of subcritical and
critical MBPRES
10.4. Functional limit theorem in the critical case
10.5. Subcritical multi-type case
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Tags: Götz Kersting, Vladimir Vatutin, Discrete Time, Branching Processes