Transience and recurrence of state-dependent branching processes with an immigration component

Abstract

We consider the following modification of an ordinary Galton–Watson branching process. If Zn = i, a positive integer, then each parent reproduces independently of one another according to the ith {P(i)k} of a countable collection of probability measures. If Zn = 0, then Zn + 1 is selected from a fixed immigration distribution. We present sufficient conditions on the means μi, the variances σ2i, and the (2 + γ)th central absolute moments β2+γ,i of the {P(i)k}'s which ensure transience of recurrence of {Zn}.