A Branching Process Allowing Immigration

Summary It is well known that in the absence of immigration, a population of like particles developing under the usual laws for branching processes either increases unboundedly with time or becomes extinct (Feller, 1957, Chapter 12; Harris, 1963, Chapter 1). If migration into the system is permitted, then it is clear that under certain conditions a proper stationary distribution for population size will exist. That this is the case has been shown by Bartlett (1956, Section 3.41) for processes in continuous time, and the present note is concerned with the same problem but considering discrete generations. Our result for the generating function of the stationary distribution of population size reduces to a form analogous to Bartlett's result when the immigration distribution is Poisson.

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