Extinction in lower Hessenberg branching processes with countably many types

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 $j\le i+1$ 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 $\mathbf{q}$ and whose maximum is the partial extinction probability vector $\tilde{\mathbf{q}}$. In the case where $\tilde{\mathbf{q}}=1$, we derive a global extinction criterion which holds under second moment conditions, and when $\tilde{\mathbf{q}}<1$ we develop necessary and sufficient conditions for $\mathbf{q}=\tilde{\mathbf{q}}$.

References in corpus (4)

• Characterization of the critical values of branching random walks on weighted graphs through infinite-type branching processes
• Extinction in lower Hessenberg branching processes with countably many types
• A unifying approach to branching processes in varying environments
• The probabilities of extinction in a branching random walk on a strip

Cited by in corpus (5)

• Extinction in lower Hessenberg branching processes with countably many types
• Branching Random Walks with Two Types of Particles on Multidimensional Lattices
• The probabilities of extinction in a branching random walk on a strip
• Boolean percolation on digraphs and random exchange processes
• Inverting the operation of conditioning a branching process on extinction