Extinction probabilities in branching processes with countably many types: a general framework

TL;DR

This paper develops a comprehensive framework for analyzing extinction probabilities in multitype Galton-Watson processes with infinitely many types, characterizing the structure and number of distinct extinction probability vectors.

Contribution

It introduces a general method to identify and classify all distinct extinction probability vectors in countably many types, extending classical results to infinite type spaces.

Findings

Characterization of fixed points of the progeny generating vector.

Conditions for strict inequalities between extinction probabilities.

Framework to determine the cardinality of the set of extinction probability vectors.

Abstract

We consider Galton-Watson branching processes with countable typeset $\mathcal{X}$. We study the vectors ${\bf q}(A)=(q_x(A))_{x\in\mathcal{X}}$ recording the conditional probabilities of extinction in subsets of types $A\subseteq \mathcal{X}$, given that the type of the initial individual is $x$. We first investigate the location of the vectors ${\bf q}(A)$ in the set of fixed points of the progeny generating vector and prove that $q_x(\{x\})$ is larger than or equal to the $x$th entry of any fixed point, whenever it is different from 1. Next, we present equivalent conditions for $q_x(A)< q_x (B)$ for any initial type $x$ and $A,B\subseteq \mathcal{X}$. Finally, we develop a general framework to characterise all \emph{distinct} extinction probability vectors, and thereby to determine whether there are finitely many, countably many, or uncountably many distinct vectors. We illustrate our results with examples, and conclude with open questions.