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 X = {0, 1, 2, . . . }, in which individuals of type i may give birth to offspring of type j ≤ 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 q and whose maximum is the partial extinction probability vectorq. In the case whereq = 1, we derive a global extinction criterion which holds under second moment conditions, and whenq < 1 we develop necessary and sufficient conditions for q =q.

The probabilities of extinction in a branching random walk on a strip

“…We begin our analysis by deriving partial and global extinction criteria for block LHBPs. These criteria extend the results in [7,Theorem 5.1]. They are based on the sequence of…”

“…Suppose x = 1. By Corollary 4.2, q = q(A 1 ) = q(A 2 ), and by [7,Corollary 3], q < 1. Note that there is partial (global) extinction in {Z n } if any only if there is partial (global) extinction in its local isomorphism {Ẑ n }.…”

“…We therefore restrict our attention to a subclass of branching processes that is more amenable to analysis. One possible subclass is the lower Hessenberg branching processes (LHBPs) considered in [7]. In these processes, which have the typeset X = {0, 1, 2, .…”

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

“…Up to the possibility of a total catastrophe, the individuals in {S k } behave independently, hence for any |s 0 | ≤ x, we have In specific cases, the extinction probability of the seed process can be easier to analyse than that of the original branching process. In [8] we consider one such subclass of branching processes called lower Hessenberg where, by building upon the results of the present section, we are able to analyse the set of fixed points of the original process and derive necessary and sufficient conditions for its almost sure global extinction.…”

“…For example, {S k } almost surely becomes extinct if and only if global and partial extinction of the original process coincide. While in the present paper our interest in the seed process remains its application to the sequence {q (k) }, we lay the foundations for a subsequent paper [8], in which properties of the seed process are exploited further, to yield, among other results, a global extinction criterion that applies to a class of branching processes referred to as lower Hessenberg.…”