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Transience and recurrence of state-dependent branching processes with an immigration component

Published online by Cambridge University Press:  01 July 2016

Joshua B. Levy
Joshua B. Levy*
Affiliation:
Georgia Institute of Technology
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Abstract

We consider the following modification of an ordinary Galton–Watson branching process. If Zn = i, a positive integer, then each parent reproduces independently of one another according to the ith {P(i)k} of a countable collection of probability measures. If Zn = 0, then Zn + 1 is selected from a fixed immigration distribution. We present sufficient conditions on the means μi, the variances σ2i, and the (2 + γ)th central absolute moments β2+γ,i of the {P(i)k}'s which ensure transience of recurrence of {Zn}.

Keywords

STATE-DEPENDENT BRANCHING PROCESSIMMIGRATIONTRANSIENCERECURRENCE
Type
Research Article
Information
Advances in Applied Probability , Volume 11 , Issue 1 , March 1979 , pp. 73 - 92
Copyright
Copyright © Applied Probability Trust 1979 

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References

1. Athreya, K. B. and Ney, P. (1972) Branching Processes. Springer-Verlag, Berlin.CrossRefGoogle Scholar
2. Esséen, C. G. (1945) Fourier analysis of distribution functions. A mathematical study of the Laplace–Gaussian law. Acta Math. 77, 1–125.CrossRefGoogle Scholar
3. Pakes, A. G. (1969) Some conditions for ergodicity and recurrence of Markov chains. Opns Res. 17, 1058–1061.CrossRefGoogle Scholar
4. Pakes, A. G. (1971) A branching process with a state-dependent immigration component. Adv. Appl. Prob. 3, 301–314.CrossRefGoogle Scholar
5. Stein, W. M. (1974) Transience and recurrence of certain state branching processes. Technical Summary Report No. 1467, Mathematics Research Center, University of Wisconsin, Madison.Google Scholar
6. Stein, W. M. (1974) Transience and recurrence of branching processes with state-dependent offspring distributions. Technical Summary Report No. 1478, Mathematics Research Center, University of Wisconsin, Madison.Google Scholar
7. Stout, W. F. (1974) Almost Sure Convergence. Academic Press, New York.Google Scholar