# Extinction in lower Hessenberg branching processes with countably many types

## TL;DR

This paper investigates extinction probabilities in a specific class of multitype branching processes with countably many types, revealing a continuum of fixed points and establishing criteria for global extinction under certain conditions.

## Contribution

It introduces the analysis of fixed points in Lower Hessenberg branching processes and provides new extinction criteria and conditions for equality of extinction probabilities.

## Key findings

- Existence of a continuum of fixed points between global and partial extinction probabilities.
- A global extinction criterion under second moment conditions when partial extinction probability is one.
- Necessary and sufficient conditions for the equality of global and partial extinction probabilities.

## Abstract

We consider a class of branching processes with countably many types which we refer to as Lower Hessenberg branching processes. These are multitype Galton-Watson processes with typeset $\mathcal{X}=\{0,1,2,\dots\}$, in which individuals of type $i$ may give birth to offspring of type $j\leq i+1$ only. For this class of processes, we study the set $S$ of fixed points of the progeny generating function. In particular, we highlight the existence of a continuum of fixed points whose minimum is the global extinction probability vector $\boldsymbol{q}$ and whose maximum is the partial extinction probability vector $\boldsymbol{\tilde{q}}$. In the case where $\boldsymbol{\tilde{q}}=\boldsymbol{1}$, we derive a global extinction criterion which holds under second moment conditions, and when $\boldsymbol{\tilde{q}}<\boldsymbol{1}$ we develop necessary and sufficient conditions for $\boldsymbol{q}=\boldsymbol{\tilde{q}}$.