Branching Process in a Varying Environment: How to Grow Like the Product of Means

Abstract

Consider a branching process $\{Z_n\}$ in a varying environment. Let $\{W_n\}$ be the natural martingale $Z_n/{\bf E}Z_n$. It converges to some random variable $W$ as $n\to\infty$. An important problem is to show that ${\bf P}(W>0)$ equals the survival probability, so that $Z_n$ is either $0$ or of the order ${\bf E}Z_n$. We find a new kind of sufficient conditions, applicable to branching processes in a random environment. An important property of our estimates is that we don't necessary assume that ${\bf E}X_{i,1}^2$ are finite for every $i$.