Defective Galton-Watson processes in a varying environment

Abstract

We study an extension of the so-called defective Galton-Watson processes obtained by allowing the offspring distribution to change over the generations. Thus, in these processes, the individuals reproduce independently of the others and in accordance to some possibly defective offspring distribution depending on the generation. Moreover, the defect $1-{\mathit{f}}_{\mathit{n}}(1)$ of the offspring distribution at generation n represents the probability that the process hits an absorbing state Δ at that generation. We focus on the asymptotic behaviour of these processes. We establish the almost sure convergence of the process to a random variable with values in ${\mathbb{N}}_0\cup \{\mathrm{\Delta }\}$ and we provide two characterisations of the duality extinction-absorption at Δ. We also state some results on the absorption time and the properties of the process conditioned upon its non-absorption, some of which require us to introduce the notion of defective branching trees in varying environment.