Critical branching processes in random environment and Cauchy domain of attraction

TL;DR

This paper studies the survival probability of critical branching processes in random environments, focusing on cases where the associated random walk oscillates and the offspring distribution is in the domain of attraction of a stable law with parameter 1, revealing asymptotic behaviors.

Contribution

It extends the analysis of critical branching processes in random environments to the case where offspring distributions are in the domain of attraction of a stable law with parameter 1, especially when the associated random walk is oscillating.

Findings

Survival probability asymptotics when $ ho=0$

Survival probability asymptotics when $ ho=1$

Behavior of processes in the domain of attraction of a stable law with parameter 1

Abstract

We are interested in the survival probability of a population modeled by a critical branching process in an i.i.d. random environment. We assume that the random walk associated with the branching process is oscillating and satisfies a Spitzer condition $\mathbf{P}(S_{n}>0)\rightarrow \rho ,\ n\rightarrow \infty $, which is a standard condition in fluctuation theory of random walks. Unlike the previously studied case $\rho \in (0,1)$, we investigate the case where the offspring distribution is in the domain of attraction of a stable law with parameter $1$, which implies that $\rho =0$ or $1$. We find the asymptotic behaviour of the survival probability of the population in these two cases.