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The Galton-Watson Process | |
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Preliminaries | |
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The Basic Setting | |
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Moments | |
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Elementary Properties of Generating Functions | |
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An Important Example | |
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Extinction Probability | |
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A First Look at Limit Theorems | |
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Motivating Remarks | |
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Ratio Theorems | |
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Conditioned Limit Laws | |
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The Exponential Limit Law for the Critical Process | |
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Finer Limit Theorems | |
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Strong Convergence in the Supercritical Case | |
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Geometric Convergence of f[subscript n](s) in the Noncritical Cases | |
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Further Ramifications | |
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Decomposition of the Supercritical Branching Process | |
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Second Order Properties of Z[subscript n]/m[superscript n] | |
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The Q-Process | |
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More on Conditioning; Limiting Diffusions | |
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Complements and Problems I | |
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Potential Theory | |
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Introduction | |
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Stationary Measures: Existence, Uniqueness, and Representation | |
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The Local Limit Theorem for the Critical Case | |
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The Local Limit Theorem for the Supercritical Case | |
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Further Properties of W; A Sharp Global Limit Law; Positivity of the Density | |
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Asymptotic Properties of Stationary Measures | |
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Green Function Behavior | |
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Harmonic Functions | |
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The Space-Time Boundary | |
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Complements and Problems II | |
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One Dimensional Continuous Time Markov Branching Processes | |
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Definition | |
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Construction | |
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Generating Functions | |
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Extinction Probability and Moments | |
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Examples: Binary Fission, Birth and Death Process | |
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The Embedded Galton-Watson Process and Applications to Moments | |
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Limit Theorems | |
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More on Generating Functions | |
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Split Times | |
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Second Order Properties | |
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Constructions Related to Poisson Processes | |
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The Embeddability Problem | |
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Complements and Problems III | |
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Age-Dependent Processes | |
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Introduction | |
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Existence and Uniqueness | |
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Comparison with Galton-Watson Process; Embedded Generation Process; Extinction Probability | |
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Renewal Theory | |
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Moments | |
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Asymptotic Behavior of F(s, t) in the Critical Case | |
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Asymptotic Behavior of F(s, t) when m[not equal]1: The Malthusian Case | |
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Asymptotic Behavior of F(s, t) when m[not equal]1: Sub-Exponential Case | |
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The Exponential Limit Law in the Critical Case | |
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The Limit Law for the Subcritical Age-Dependent Process | |
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Limit Theorems for the Supercritical Case | |
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Complements and Problems IV | |
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Multi-Type Branching Processes | |
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Introduction and Definitions | |
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Moments and the Frobenius Theorem | |
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Extinction Probability and Transience | |
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Limit Theorems for the Subcritical Case | |
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Limit Theorems for the Critical Case | |
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The Supercritical Case and Geometric Growth | |
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The Continuous Time, Multitype Markov Case | |
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Linear Functionals of Supercritical Processes | |
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Embedding of Urn Schemes into Continuous Time Markov Branching Processes | |
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The Multitype Age-Dependent Process | |
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Complements and Problems V | |
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Special Processes | |
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A One Dimensional Branching Random Walk | |
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Cascades; Distributions of Generations | |
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Branching Diffusions | |
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Martingale Methods | |
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Branching Processes with Random Environments | |
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Continuous State Branching Processes | |
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Immigration | |
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Instability | |
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Complements and Problems VI | |
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Bibliography | |
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List of Symbols | |
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Author Index | |
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Subject Index | |