Hostname: page-component-6bb9c88b65-znhjv Total loading time: 0 Render date: 2025-07-25T23:49:52.687Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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

Information

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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