Branching processes with immigration

Abstract

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 or α > 1, respectively. Unless stated specifically to the contrary, we assume that the following condition holds.