Nearly critical Galton--Watson processes

Abstract

We investigate Galton--Watson processes in varying environment, for which ${\bar{f}}_n\uparrow 1$ and ${\sum }_{n=1}^{\infty }(1-{\bar{f}}_n)=\infty$, where ${\bar{f}}_n$ stands for the offspring mean in generation $n$. Since the process dies out almost surely, to obtain nontrivial limit we consider two scenarios: conditioning on non-extinction, or adding immigration. In both cases we show that the process converges in distribution without normalization to a nondegenerate compound-Poisson limit law. The proofs rely on the shape function technique, worked out by Kersting (2020).