Extinction probabilities of branching processes with countably infinitely many types

TL;DR

This paper introduces iterative methods to compute extinction probabilities in complex branching processes with infinitely many types, linking matrix norms to extinction criteria and providing conditions for almost sure extinction despite population explosion.

Contribution

It develops new iterative algorithms for extinction probability calculation in infinite-type branching processes and explores their theoretical properties and implications.

Findings

Methods effectively compute extinction probabilities for infinite-type processes.

Connection established between matrix norms and extinction criteria.

Identifies conditions for almost sure extinction despite population growth.

Abstract

We present two iterative methods for computing the global and partial extinction probability vectors for Galton-Watson processes with countably infinitely many types. The probabilistic interpretation of these methods involves truncated Galton-Watson processes with finite sets of types and modified progeny generating functions. In addition, we discuss the connection of the convergence norm of the mean progeny matrix with extinction criteria. Finally, we give a sufficient condition for a population to become extinct almost surely even though its population size explodes on the average, which is impossible in a branching process with finitely many types. We conclude with some numerical illustrations for our algorithmic methods.