Branching Processes

ISBN-10: 0486434745

ISBN-13: 9780486434742

Edition: 2004

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Description: A unified treatment of the limit theory of branching processes, this volume focuses on basics and is appropriate for graduate and advanced undergraduate students. The authors cover basic Galton-Watson process, potential theory, one dimensional continuous time Markov branching processes, age-dependent processes, multi-type branching processes, and special processes. Exercises. 1972 edition.

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Book details

List price: $17.95
Copyright year: 2004
Publisher: Dover Publications, Incorporated
Publication date: 3/19/2004
Binding: Paperback
Pages: 304
Size: 5.25" wide x 8.25" long x 0.50" tall
Weight: 0.682
Language: English

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