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

Consider a branching process $\{Z_n\}$ in a varying environment. Let $\{W_n\}$ be the natural martingale $Z_n\mathrm{/}\mathbf{E}{Z}_n$. It converges to some random variable $W$ as $n\to \mathrm{\infty }$. An important problem is to show that $\mathbf{P}(W>0)$ equals the survival probability, so that $Z_n$ is either $0$ or of the order $\mathbf{E}{Z}_n$. 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 $\mathbf{E}{X}_{i,1}^2$ are finite for every $i$.