The Survival Probability of a Critical Branching Process in a Random Environment

Abstract

In this paper we determine the asymptotic behavior of the survival probability of a critical branching process in a random environment. In the special case of independent identically distributed geometric offspring distributions, and the somewhat more general case of offspring distributions with linear fractional generating functions, Kozlov proved that, as $n\to\infty$, the probability of nonextinction at generation n is proportional to $n^{-1/2}$. We establish Kozlov's asymptotic for general independent identically distributed offspring distributions.