Branching Processes

ISBN-13: 9780817631222 / Angielski / Miękka / 1983 / 461 str.

Asmussen;Hering

Opis

Branching processes form one of the classical fields of applied probability and are still an active area of research. The field has by now grown so large and diverse that a complete and unified treat- ment is hardly possible anymore, let alone in one volume. So, our aim here has been to single out some of the more recent developments and to present them with sufficient background material to obtain a largely self-contained treatment intended to supplement previous mo- nographs rather than to overlap them. The body of the text is divided into four parts, each of its own flavor. Part A is a short introduction, stressing examples and applications. In Part B we give a self-contained and up-to-date pre- sentation of the classical limit theory of simple branching processes, viz. the Gal ton-Watson ( Bienayme-G-W) process and i ts continuous time analogue. Part C deals with the limit theory of Il arkov branching processes with a general set of types under conditions tailored to (multigroup) branching diffusions on bounded domains, a setting which also covers the ordinary multitype case. Whereas the point of view in Parts A and B is quite pedagogical, the aim of Part C is to treat a large subfield to the highest degree of generality and completeness possi"ble. Thus the exposition there is at times quite technical.

Spis treści

I: Branching phenomena and models.- II: The Galton-Watson process: Probabilistic methods.- III: The Galton-Watson process: Analytic methods.- IV: Continuous time Markov branching processes.- V: Foundations.- VI: Limit theory for subcritical and critical processes.- VII: Basic limit theory for supercritical processes.- VIII: More on the limiting behaviour of linear functionals.- IX: Unbounded domains.- X: Generalized age-dependence and random characteristics.- XI: Two-sex models.- 1. The conditional Borel-Cantelli lemma.- 2. Martingale convergence theorems.- 3. Uniform integrability.- 4. Series with independent terms.- 5. Summation by parts.- 6. Maximal inequalities.- 7. Results related to the LIL.- 8. The martingale CLT.- 9. The Croft-Kingman lemma.- 10. Results related to the LNN.- 11. A result of Anscombe-Renyi type.- 12. A weak LLN rate of convergence result.- 13. Slowly or regularly varying functions.- 14. Tauberian theorems.- References.