The purpose of this book is to give a unified treatment of the limit theory of branching processes. Since the publication of the important book of T E. Harris (Theory of Branching Processes, Springer, 1963) the subject has developed and matured significantly. Many of the classical limit laws are now known in their sharpest form, and there are new proofs that give insight into the results. Our work deals primarily with this decade, and thus has very little overlap with that of Harris. Only enough material is repeated to make the treatment essentially self-contained. For example, certain foundational questions on the construction of processes, to which we have nothing new to add, are not developed. There is a natural classification of branching processes according to their criticality condition, their time parameter, the single or multi-type particle cases, the Markovian or non-Markovian character of the pro­ cess, etc. We have tried to avoid the rather uneconomical and un­ enlightening approach of treating these categories independently, and by a series of similar but increasingly complicated techniques. The basic Galton-Watson process is developed in great detail in Chapters I and II.

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I. The Galton-Watson Process.- A. Preliminaries.- 1. The Basic Setting.- 2. Moments.- 3. Elementary Properties of Generating Functions.- 4. An Important Example.- 5. Extinction Probability.- B. A First Look at Limit Theorems.- 6. Motivating Remarks.- 7. Ratio Theorems.- 8. Conditioned Limit Laws.- 9. The Exponential Limit Law for the Critical Process.- C. Finer Limit Theorems.- 10. Strong Convergence in the Supercritical Case.- 11. Geometric Convergence of fn(s) in the Noncritical Cases.- D. Further Ramifications.- 12. Decomposition of the Supercritical Branching Process.- 13. Second Order Properties of Zn/mn.- 14. The Q-Process.- 15. More on Conditioning; Limiting Diffusions.- 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.