Branching processes in stationary random environment: The extinction problem revisited

Abstract

A classical result by Athreya and Karlin states that a supercritical Galton-Watson process in stationary ergodic environment $\textbf{f}=(f_{0},f_{1},{\ldots})$ (these are the random generating functions of the successively picked offspring distributions) has a positive chance of survival $1-q(\textbf{f})$ for almost all realizations of f provided that $\mathbb{E}\log(1-f_{0}(0))>-\infty$. While in some cases like when $f_{0},f_{1},{\ldots}$ are i.i.d., this last condition together with supercriticality, viz. $\mathbb{E}\log f_{0}'(1)>0$, is actually equivalent to $q(\textbf{f}) <1$ a.s., there are others where it is not. This is demonstrated by giving a rather simple counterexample which in turn draws on the main result of this paper. The latter is intended to shed further light on the relation between $\mathbb{E}\log(1-f_{0}(0))>-\infty$ and the almost sure noncertain extinction property, the most interesting outcome being that, if $\mathbb{E}\log f_{0}'(1)$ is also finite, then $q(\textbf{f}) <1$ a.s. holds iff $\mathbb{E}\log\left(\frac{1-f_{0}\circ{\ldots}\circ f_{T}(0)}{1-f_{1}\circ{\ldots}\circ f_{T}(0)}\right)>-\infty$ for some random time T. The use of random times in connection with the stationary environment f will lead us quite naturally to the use of Palm-duality theory in some of our arguments.

Mathematics Subject Classification (2000): 60J80