Defective Galton-Watson processes in a varying environment

TL;DR

This paper investigates a generalized defective Galton-Watson process with varying offspring distributions, analyzing its long-term behavior, extinction probabilities, and absorption characteristics, extending classical branching process theory to more complex, environment-dependent scenarios.

Contribution

It introduces a novel framework for defective Galton-Watson processes in a changing environment, providing convergence results and duality characterizations not previously established.

Findings

Almost sure convergence to a random variable or absorption state

Duality between extinction and absorption at the defect point

Results on absorption time and conditioned process properties

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 $1-f_n(1)$ of the offspring distribution at generation $n$ represents the probability that the process hits an absorbing state $\Delta$ 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 $\mathbb{N}_0\cup\{\Delta\}$ and we provide two characterisations of the duality extinction-absorption at $\Delta$. 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.