Branching processes with immigration

Consider a branching process in which each individual reproduces independently of all others and has probability a j ( j = 0, 1, ···) of giving rise to j progeny in the following generation, and in which there is an independent immigration component where, with probability b j ( j = 0, 1, ···) j objects enter the population at each generation. Then letting X n ( n = 0, 1, ···) be the population size of the nth generation, it is known (Heathcote (1965), (1966)) that { X n } defines a Markov chain on the non-negative integers and it is called a branching process with immigration (b.p.i.). We shall call the process sub-critical or super-critical according as the mean number of offspring of an individual, , satisfies α 1, respectively. Unless stated specifically to the contrary, we assume that the following condition holds.

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