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 vector q~. In the case where q~=1, we derive a global extinction criterion which holds under second moment conditions, and when q~<1 w..