Nearly critical Galton--Watson processes

TL;DR

This paper studies nearly critical Galton--Watson processes with varying environments, showing they converge to a compound-Poisson distribution under certain conditions, using the shape function technique.

Contribution

It introduces a detailed analysis of nearly critical Galton--Watson processes with varying environments, extending understanding of their limit laws.

Findings

Processes converge to a compound-Poisson distribution when conditioned on non-extinction or with immigration.

The convergence occurs without normalization in the specified setting.

The shape function technique is effectively applied to analyze these 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).