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A note on the probability of extinction in a class of population-size-dependent Galton-Watson processes

Published online by Cambridge University Press:  14 July 2016

R. Höpfner
R. Höpfner*
Affiliation:
Albert-Ludwigs-Universität, Freiburg
*
∗ Postal address: Institut für Mathematische Stochastik, Albert-Ludwigs-Universität, Hebelstr. 27, D-7800 Freiburg, West Germany.
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Abstract

In a class of population-size-dependent Galton-Watson processes where extinction does not occur with probability 1 we describe the rate of decay of qi (the probability that the process starting from i ancestors will become extinct) as the number i of ancestors increases.

The results are related to the asymptotic behavior of the Green's function of the critical Galton-Watson process with immigration.

Keywords

STATE DEPENDENCEIMMIGRATIONEXTINCTIONGREEN'S FUNCTION
Type
Short Communications
Information
Journal of Applied Probability , Volume 22 , Issue 4 , December 1985 , pp. 920 - 925
Copyright
Copyright © Applied Probability Trust 1985 

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References

[1] Fujimagari, T. (1976) Controlled Galton-Watson process and its asymptotic behavior. Kodai Math. Sem. Rep. 27, 11–18.Google Scholar
[2] Galambos, J. and Seneta, E. (1973) Regularly varying sequences. Proc. Amer. Math. Soc. 41, 110–116.Google Scholar
[3] Höpfner, R. (1983) über einige Klassen von zustandsabhängigen Galton-Watson Prozessen. Dissertation, Fachbereich Mathematik, Johannes-Gutenberg-Universität Mainz.Google Scholar
[4] Höpfner, R. (1985) On some classes of population-size-dependent branching processes. J. Appl. Prob. 22, 25–36.Google Scholar
[5] Ivanoff, B. G. and Seneta, E. (1985) The critical branching processes with immigration stopped at zero. J. Appl. Prob. 22, 223–227.CrossRefGoogle Scholar
[6] Klebaner, F. C. (1983) Population-size-dependent branching process with linear rate of growth. J. Appl. Prob. 20, 242–250.Google Scholar
[7] Klebaner, F. C. (1984) On population-size-dependent branching processes. Adv. Appl. Prob. 16, 30–55.Google Scholar
[8] Klebaner, F. C. (1984) Geometric rate of growth in population-size-dependent Galton-Watson processes. J. Appl. Prob. 21, 40–49.Google Scholar
[9] Klebaner, F. C. (1985) A limit theorem for population-size-dependent branching processes. J. Appl. Prob. 22, 48–57.Google Scholar
[10] Küster, P. (1985) Asymptotic growth of controlled Galton-Watson processes. Ann. Prob. Google Scholar
[11] Levy, J. B. (1979) Transience and recurrence of state-dependent branching processes with an immigration component. Adv. Appl. Prob. 11, 73–92.Google Scholar
[12] Pakes, A. G. (1971) On the critical Galton-Watson process with immigration. J. Austral. Math. Soc. 12, 476–482.CrossRefGoogle Scholar
[13] Pakes, A. G. (1972) Further results on the critical Galton-Watson process with immigration. J. Austral. Math. Soc. 13, 277–290.Google Scholar
[14] Roi, L. D. (1975) State-Dependent Branching Processes. Thesis, Purdue University.Google Scholar
[15] Seneta, H. and Tavaré, S. (1983) A note on models using the branching process with immigration stopped at zero. J. Appl. Prob. 20, 11–18.Google Scholar
[16] Zubkov, A. M. (1972) Life periods of a branching process with immigration. Theory Prob. Appl. 17, 174–183.Google Scholar