Discrete time branching processes in random environment. Volume 1

Discrete time branching processes in random environment Volume 1 Volume 1 /

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...

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Detalles Bibliográficos
Otros Autores: Kersting, Gotz, author (author), Vatutin, Vladimir, author
Formato: Libro electrónico
Idioma:Inglés
Publicado: London, England ; Hoboken, New Jersey : ISTE 2017.
Edición:1st edition
Colección:Mathematics and statistics series (ISTE)
THEi Wiley ebooks.
Materias:
Branching processes.
Ver en Biblioteca Universitat Ramon Llull:https://discovery.url.edu/permalink/34CSUC_URL/1im36ta/alma991009630637906719
Tabla de Contenidos:
  • 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 &lt
  • t &lt
  • 1
  • 6.4.3. The number of particles at moment τ(n) ≤ nt.
  • 6.4.4. The number of particles at moment τ(n) &gt
  • 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.

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