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Further results 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 UniversityMelbourne, 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, 1, …) of giving rise to j progeny in the following generation, and in which there is an independent immigration component where bj, (j = 0, 1, …) is the probability that j individuals enter the population at each generation. Defining Xn (n = 0, 1, …) to be the population size at the n- th generation, it is known that {Xn} defines a Markov chain on the non-negative integers.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 13 , Issue 3 , May 1972 , pp. 277 - 290
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Bharucha-Reid, A. T., Elements of the Theory of Markov Processes and their Applications (New York, McGraw-Hill, 1960).Google Scholar
[2]Feller, W., An Introduction to Probability Theory and its Applications. Vol. II (New York, Wiley, 1966).Google Scholar
[3]Harris, T. E., The Theory of Branching Processes (Berlin, Springer-Verlag, 1963).Google Scholar
[4]Karlin, S. and McGregor, J., ‘Spectral representation of branching processes’, Zeit. Wahrsch. 5 (1966), 34–54.Google Scholar
[5]Kesten, H., Ney, P. and Spitzer, F., ‘The Galton-Watson process with mean one and finite variance’, Teor. Veroyatnost. i Primenen. 11 (1966), 579–611.Google Scholar
[6]Pakes, A. G., ‘On the critical Galton-Watson process with immigration’, J. Aust. Math. Soc. 12 (1971), 476–482.Added in proof: The density function in the statement of Theorem 3 (page 480) is incorrect, substitute γ for β.Google Scholar
[7]Pakes, A. G., ‘Branching processes with immigration’, J. Appl. Prob. 8 (1971), 32–42.Google Scholar
[8]Seneta, E., ‘On asymptotic properties of subcritical branching processes’, J. Aust. Math. Soc. 8 (1968), 671–682.Google Scholar
[9]Seneta, E., ‘Functional equations and the Galton-Watson process’, Adv. Appl. Prob. 1 (1969), 1–42.Google Scholar
[10]Titchmarsh, E. C., The Theory of Functions. (O.U.P., 1939).Google Scholar