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

TL;DR

This paper investigates branching processes in varying environments, establishing new conditions for survival probability and growth behavior without requiring finite second moments.

Contribution

It introduces novel sufficient conditions for survival and growth in branching processes in random environments, relaxing the need for finite second moments.

Findings

Established conditions equate survival probability with the positivity of the martingale limit.

Proved that the process either dies out or grows at the rate of the mean under new conditions.

Applicable to processes with potentially infinite second moments.

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 $i$.