Multitype branching processes evolving in i.i.d. random environment: probability of survival for the critical case

TL;DR

This paper analyzes the survival probability of critical multitype branching processes in i.i.d. random environments, showing it decays as a constant times n^{-1/2} under general conditions, extending previous fractional linear cases.

Contribution

It generalizes the asymptotic survival probability results to broader offspring generating functions beyond fractional linear forms.

Findings

Survival probability behaves as c_i n^{-1/2} for large n.

Asymptotic behavior holds under general offspring generating functions.

Extends previous results from fractional linear to more general cases.

Abstract

Using the annealed approach we investigate the asymptotic behavior of the survival probability of a critical multitype branching process evolving in i.i.d. random environment. We show under rather general assumptions on the form of the offspring generating functions of particles that the probability of survival up to generation $n$ of the process initiated at moment zero by a single particle of type $i$ is equivalent to $c_{i}n^{-1/2}$ for large $n,$ where $c_{i}$ is a positive constant. Earlier such asymptotic representation was known only for the case of the fractional linear offspring generating functions.