Galton--Watson branching processes in a random environment.~I: limit theorems

V. A. Vatutin ; E. E. D'yakonova

Teoriâ veroâtnostej i ee primeneniâ, Tome 48 (2003) no. 2, pp. 274-300

Résumé

Let $Z_n$ be the number of individuals at time $n$ in a branching process in a random environment generated by independent identically distributed random probability generating functions $f_0(s),f_1(s),\dots,f_n(s),\dots$ . Let $$ X_i=\log f_{i-1}'(1),\qquad i=0,1,2,\dots; \qquad S_0=0,\quad S_n=X_1+\dots+X_n,\qquad n\ge 1. $$ It is shown that if $Z_n$ is, in a sense, “critical,” then there exists a limit in distribution $$ \lim_{n\to\infty}\exp\Bigl\{-\min_{0\le j\le n}S_j\Bigr\}\,\mathbf{P}\{Z_n>0\mid f_0,\dots,f_{n-1}\}=\zeta, $$ where $\zeta$ is a proper random variable positive with probability 1. In addition, it is shown that for a “typical” realization of the environment the number of individuals $Z_n$ given $\{Z_n>0\}$ grows as $\exp\{S_n-\min_{0\le j\le n}S_j\}$ (up to a positive finite random multiplier).

Détail

Keywords: branching processes in random environment, survival probability, critical branching process, random walks, harmonic functions.
Mots-clés : stable distributions