The Stationary Distribution of a Branching Process Allowing Immigration: A Remark on the Critical Case

Summary For a branching process in which the individuals reproduce from generation to generation as in a Galton-Watson process, and are also subject to an immigration component at each generation, it is known that a stationary-limiting distribution sometimes exists. In particular, this is so if the mean number of offspring per individual is less than unity and the immigration distribution is reasonably behaved. We show that such a distribution may exist even when the mean number of offspring is unity, under special circumstances for which we give sufficient conditions.

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