On the critical Galton-Watson process with immigration

Consider a Galton-Watson process in which each individual reproduces independently of all others and has probability a j ( j = 0, l, …) of giving rise to j progeny in the following generation and in which there is an independent immigration component where b j ( j = 0, l, …) is the probability that j individuals enter the population at each generation. Then letting X n ( n = 0, l, …) be the population size of the n-th generation, it is known (Heathcote [4], [51]) that { X n } defines a Markov chain on the non-negative integers. Unless otherwise stated, we shall consider only those offspring and immigration distributions that make the Markov chain { X n } irreducible and aperiodic.

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