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by Vladimir Vatutin, Götz Kersting
Discrete Time Branching Processes in Random Environment
Cover
Title
Copyright
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
1.5. 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
4.5. 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
6.5. Discrete limit distributions
6.6. Notes
7 Weakly Subcritical BPREs
7.1. Introduction
7.2. The probability measures and
7.3. Proof of theorems
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
9.4. Notes
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
Appendix
A.1. Examples of slowly varying functions
Bibliography
Index
End User License Agreement
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Prev
Previous Chapter
Cover
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Next Chapter
Title
Table of Contents
Cover
Title
Copyright
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
1.5. 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
4.5. 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 Z
n
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
6.5. Discrete limit distributions
6.6. Notes
7 Weakly Subcritical BPREs
7.1. Introduction
7.2. The probability measures
and
7.3. Proof of theorems
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
9.4. Notes
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
Appendix
A.1. Examples of slowly varying functions
Bibliography
Index
End User License Agreement
List of Illustrations
1 Branching Processes in Varying Environment
Figure 1.1. A rooted tree
Figure 1.2. A Geiger tree
2 Branching Processes in Random Environment
Figure 2.1. Decay rates for the quenched and annealed case
Figure 2.2. Three subcritical regimes
3 Large Deviations for BPREs
Figure 3.1. The rate function Γ
Figure 3.2. Caricatures of different types of paths
4 Properties of Random Walks
Figure 4.1. A random walk path {S
0
, … S
n
} and its permutation
Figure 4.2. The shift of ladder epochs
8 Intermediate Subcritical BPREs
Figure 8.1. A simulation in the intermediate case
Guide
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Table of Contents
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