The supercritical Galton-Watson process in varying environments

Abstract

Let { Z n } 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 { W > 0} need not coincide with { Z n → ∞}. 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 { Z n EZ n } 1 n → 1 on { Z n → ∞}.