Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T17:26:56.257Z Has data issue: false hasContentIssue false
  • English
  • Français

On the asymptotic behaviour of the extinction time of the simple branching process

Published online by Cambridge University Press:  01 July 2016

Anthony G. Pakes
Anthony G. Pakes*
Affiliation:
The University of Western Australia
*
∗Postal address: Department of Mathematics, The University of Western Australia, Nedlands, WA 6009, Australia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The time to extinction of a subcritical Galton–Watson branching process and the time of last mutation of its infinite-alleles version are maxima of independent random variables having an upper tail of geometric type, and hence they are not attracted to any extreme value distribution. It is shown that Anderson's asymptotic results for maxima of discrete variates are applicable, and this rectifies a false assertion made in respect to the infinite-alleles simple branching process.

Keywords

MAXIMAEXTREME VALUE DISTRIBUTION
Type
Letters to the Editor
Information
Advances in Applied Probability , Volume 21 , Issue 2 , June 1989 , pp. 470 - 472
Copyright
Copyright © Applied Probability Trust 1989 

References

Anderson, C. W. (1970) Extreme value theory for a class of discrete distributions with applications to some stochastic processes. J. Appl. Prob. 7, 99–113.Google Scholar
Anderson, C. W. (1980) Local limit theorems for the maxima of discrete random variables. Math. Proc. Camb. Phil. Soc. 88, 161–165.CrossRefGoogle Scholar
Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Griffiths, R. C. and Pakes, A. G. (1988) An infinite-alleles version of the simple branching process. Adv. Appl. Prob. 20, 489–524.Google Scholar
Pakes, A. G. (1989) Asymptotic results for the extinction time of Markov branching processes allowing emigration, I. Random walk decrements. Adv. Appl. Prob. 21, 243–269.CrossRefGoogle Scholar
Resnick, S. I. (1987) Extreme Values, Regular Variation, and Point Processes. Springer-Verlag, New York.CrossRefGoogle Scholar
Seneta, E. (1974) Regularly varying functions in the theory of simple branching processes. Adv. Appl. Prob. 6, 408–420.Google Scholar
Slack, R. S. (1968) A branching process with mean one and possibly infinite variance. Z. Wahrscheinlichkeitsth. 9, 139–145.Google Scholar