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On the critical Galton-Watson process with immigration

Published online by Cambridge University Press:  09 April 2009

A. G. Pakes
A. G. Pakes
Affiliation:
Department of Mathematics Monash University Melbourne, Australia
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Consider a Galton-Watson process in which each individual reproduces independently of all others and has probability aj (j = 0, l, …) of giving rise to j progeny in the following generation and in which there is an independent immigration component where bj (j = 0, l, …) is the probability that j individuals enter the population at each generation. Then letting Xn (n = 0, l, …) be the population size of the n-th generation, it is known (Heathcote [4], [51]) that {Xn} 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 {Xn} irreducible and aperiodic.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 12 , Issue 4 , November 1971 , pp. 476 - 482
Copyright
Copyright © Australian Mathematical Society 1971

References

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