Strong survival and extinction for branching random walks via new order for generating functions

Abstract We consider general discrete-time multitype branching processes on a countable set X . According to these processes, a particle of type $x\in X$ generates a random number of children and chooses their type in X , not necessarily independently nor with the same law for different parent types. We introduce a new type of stochastic ordering of multitype branching processes, generalising the germ order introduced by Hutchcroft, which relies on the generating function of the process. We prove that given two multitype branching processes with laws ${\boldsymbol{\mu}}$ and ${\boldsymbol{\nu}}$ respectively, with ${\boldsymbol{\mu}}\ge{\boldsymbol{\nu}}$ , then in every set where there is survival according to ${\boldsymbol{\nu}}$ , there is also survival according to ${\boldsymbol{\mu}}$ . Moreover, in every set where there is strong survival according to ${\boldsymbol{\nu}}$ , there is also strong survival according to ${\boldsymbol{\mu}}$ , provided that the supremum of the global extinction probabilities for the ${\boldsymbol{\nu}}$ process, taken over all starting points x , is strictly smaller than 1. New conditions for survival and strong survival for inhomogeneous multitype branching processes are provided. We also extend a result of Moyal which claims that, under some conditions, the global extinction probability for a multitype branching process is the only fixed point of its generating function, whose supremum over all starting coordinates may be smaller than 1.