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

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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 μ i , the variances σ2 i , and the (2 + γ)th central absolute moments β2+γ,i of the {P (i) k }'s which ensure transience of recurrence of {Zn }.