The probabilities of extinction in a branching random walk on a strip

TL;DR

This paper studies extinction probabilities in multitype Galton-Watson processes with infinite types, providing criteria, iterative computation methods, and analyzing the fixed points related to subset-specific extinction probabilities.

Contribution

It introduces new criteria and iterative methods for calculating subset-specific extinction probabilities in complex multitype branching processes.

Findings

Derived partial and global extinction criteria

Developed an iterative method for computing extinction probabilities

Analyzed the fixed points of the progeny generating vector

Abstract

We consider a class of multitype Galton-Watson branching processes with a countably infinite type set $\mathcal{X}_d$ whose mean progeny matrices have a block lower Hessenberg form. For these processes, the probability $\boldsymbol{q}(A)$ of extinction in subsets of types $A\subseteq \mathcal{X}_d$ may differ from the global extinction probability $\boldsymbol{q}$ and the partial extinction probability $\tilde{\boldsymbol{q}}$. After deriving partial and global extinction criteria, we develop conditions for $\boldsymbol{q}<\boldsymbol{q}(A)<\tilde{\boldsymbol{q}}$. We then present an iterative method to compute the vector $\boldsymbol{q}(A)$ for any set $A$. Finally, we investigate the location of the vectors $\boldsymbol{q}(A)$ in the set of fixed points of the progeny generating vector.