The Supercritical Galton-Watson process in varying environments

# The Supercritical Galton-Watson process in varying environments

Stoc. Proc. Appl. (1992) 42, 39-47.

## Abstract

Let $\left\{ Z_{n}\right\}$ be a supercritical Galton-Watson process in varying environments. It is known that $Z_{n}$ when normed by its mean $EZ_{n}$ converges almost surely to a finite random variable $W$. It is possible, however, for such a process to exhibit more than one rate of growth so that in particular $\left\{ W>0 \right\}$ need not coincide with $\left\{ Z_{n} \rightarrow \infty \right\}$. Here a natural sufficient condition is given which ensures that this cannot happen. Under a weaker condition it is shown that the possible rates of growth cannot differ very much in that $\left\{ Z_{n}/EZ_{n}\right\}^{1/n} \rightarrow 1$ on $\left\{ Z_{n}\rightarrow \infty \right\}$.

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