Discrete Time Branching Processes in Random Environment.
Kersting, Götz.
Discrete Time Branching Processes in Random Environment. - 1st ed. - 1 online resource (311 pages)
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 < -- t < -- 1 -- 6.4.3. The number of particles at moment τ(n) ≤ nt. 6.4.4. The number of particles at moment τ(n) > -- nt -- 6.5. Discrete limit distributions -- 6.6. Notes -- 7. Weakly Subcritical BPREs -- 7.1. Introduction -- 7.2. The probability measures P+ 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 -- 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 -- 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 -- Other titles from iSTE in Mathematics and Statistics -- EULA.
9781119473558
Branching processes.
Electronic books.
QA274.76 .K477 2017
Discrete Time Branching Processes in Random Environment. - 1st ed. - 1 online resource (311 pages)
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 < -- t < -- 1 -- 6.4.3. The number of particles at moment τ(n) ≤ nt. 6.4.4. The number of particles at moment τ(n) > -- nt -- 6.5. Discrete limit distributions -- 6.6. Notes -- 7. Weakly Subcritical BPREs -- 7.1. Introduction -- 7.2. The probability measures P+ 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 -- 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 -- 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 -- Other titles from iSTE in Mathematics and Statistics -- EULA.
9781119473558
Branching processes.
Electronic books.
QA274.76 .K477 2017