Kevei Péter (Szeged): Nearly critical Galton--Watson processes

We investigate Galton--Watson processes in varying environment, where the offspring distribution is subcritical, and its mean tends to 1. The process with Bernoulli offspring distribution was introduced by Ispány, Györfi, Pap and Varga in 2007. 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. We also discuss functional limit theorems. The proofs rely on the shape function technique, worked out recently by Kersting.

The talk is based on ongoing joint work with Kata Kubatovics (Szeged).