Author | Athreya, Krishna B. author |
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Title | Branching Processes [electronic resource] / by Krishna B. Athreya, Peter E. Ney |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg, 1972 |
Connect to | http://dx.doi.org/10.1007/978-3-642-65371-1 |
Descript | XII, 288 p. online resource |
I. The Galton-Watson Process -- A. Preliminaries -- B. A First Look at Limit Theorems -- C. Finer Limit Theorems -- D. Further Ramifications -- Complements and Problems I -- II. Potential Theory -- 1. Introduction -- 2. Stationary Measures: Existence, Uniqueness, and Representation -- 3. The Local Limit Theorem for the Critical Case -- 4. The Local Limit Theorem for the Supercritical Case -- 5. Further Properties of W; A Sharp Global Limit Law; Positivity of the Density -- 6. Asymptotic Properties of Stationary Measures -- 7. Green Function Behavior -- 8. Harmonic Functions -- 9. The Space-Time Boundary -- Complements and Problems II -- III. One Dimensional Continuous Time Markov Branching Processes -- 1. Definition -- 2. Construction -- 3. Generating Functions -- 4. Extinction Probability and Moments -- 5. Examples: Binary Fission, Birth and Death Process -- 6. The Embedded Galton-Watson Process and Applications to Moments -- 7. Limit Theorems -- 8. More on Generating Functions -- 9. Split Times -- 10. Second Order Properties -- 11. Constructions Related to Poisson Processes -- 12. The Embeddability Problem -- Complements and Problems III -- IV. Age-Dependent Processes -- 1. Introduction -- 2. Existence and Uniqueness -- 3. Comparison with Galton-Watson Process; Embedded Generation Process; Extinction Probability -- 4. Renewal Theory -- 5. Moments -- 6. Asymptotic Behavior of F(s, t) in the Critical Case -- 7. Asymptotic Behavior of F(s, t) when m?1: The Malthusian Case -- 8. Asymptotic Behavior of F(s, t) when m?1: Sub-Exponential Case -- 9. The Exponential Limit Law in the Critical Case -- 10. The Limit Law for the Subcritical Age-Dependent Process -- 11. Limit Theorems for the Supercritical Case -- Complements and Problems IV -- V. Multi-Type Branching Processes -- 1. Introduction and Definitions -- 2. Moments and the Frobenius Theorem -- 3. Extinction Probability and Transience -- 4. Limit Theorems for the Subcritical Case -- 5. Limit Theorems for the Critical Case -- 6. The Supercritical Case and Geometric Growth -- 7. The Continuous Time, Multitype Markov Case -- 8. Linear Functionals of Supercritical Processes -- 9. Embedding of Urn Schemes into Continuous Time Markov Branching Processes -- 10. The Multitype Age-Dependent Process -- Complements and Problems V -- VI. Special Processes -- 1. A One Dimensional Branching Random Walk -- 2. Cascades; Distributions of Generations -- 3. Branching Diffusions -- 4. Martingale Methods -- 5. Branching Processes with Random Environments -- 6. Continuous State Branching Processes -- 7. Immigration -- 8. Instability -- Complements and Problems VI -- List of Symbols -- Author Index