A pathwise iterative approach to the extinction of branching processes with countably many types

TL;DR

This paper introduces a pathwise iterative method to analyze extinction probabilities in complex branching processes with infinitely many types, providing new computational techniques for these probabilistic models.

Contribution

It develops a novel pathwise approach and iterative algorithms for calculating extinction probabilities in infinite-type Galton-Watson processes.

Findings

Convergence of extinction probability vectors under certain conditions

Construction of finite truncated and augmented processes

Effective iterative methods for global extinction probability computation

Abstract

We consider the extinction events of Galton-Watson processes with countably infinitely many types. In particular, we construct truncated and augmented Galton-Watson processes with finite but increasing sets of types. A pathwise approach is then used to show that, under some sufficient conditions, the corresponding sequence of extinction probability vectors converges to the global extinction probability vector of the Galton-Watson processes with countably infinitely many types. This gives rise to a number of iterative methods for the computation of the global extinction probability vector.