The Supercritical Galton-Watson process in varying environments
#
The Supercritical Galton-Watson process in varying environments

by J.D.Biggins and
J.C. D'Souza

*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 $\left\{ Z_{n}\rightarrow
\infty \right\}$.

## Availability

postscript file
dvi file

See also the closely related paper

Other publications by J.D. Biggins
Return to my home page