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 element of upper X x ∈ X $x\in X$ 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 bold italic mu μ ${\boldsymbol{\mu}}$ and bold italic nu ν ${\boldsymbol{\nu}}$ respectively, with bold italic mu greater than or equals bold italic nu μ ≥ ν ${\boldsymbol{\mu}}\ge{\boldsymbol{\nu}}$ , then in every set where there is survival according to bold italic nu ν ${\boldsymbol{\nu}}$ , there is also survival according to bold italic mu μ ${\boldsymbol{\mu}}$ . Moreover, in every set where there is strong survival according to bold italic nu ν ${\boldsymbol{\nu}}$ , there is also strong survival according to bold italic mu μ