Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-08T07:29:50.315Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on the asymptotic behaviour of the extinction probability in supercritical population-size-dependent branching processes with independent and identically distributed random environments

Part of: Markov processes Special processes

Published online by Cambridge University Press:  14 July 2016

Lu Zhunwei  and
Peter Jagers
Lu Zhunwei*
Affiliation:
Taiyuan University of Technology
Peter Jagers*
Affiliation:
Chalmers University of Technology and Göteborg University
*
∗ Postal address: Department of Mathematics, Taiyuan University of Technology, 030024, Taiyuan, Shanxi Province, P. R. China. Email address: [email protected]
∗∗ Postal address: Department of Mathematical Statistics, Chalmers University of Technology and Göteborg University, SE-412 96, Göteborg, Sweden. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In supercritical population-size-dependent branching processes with independent and identically distributed random environments, it is shown that under certain regularity conditions there exist constants 0 < α 1 ≤α 0 < + ∞ and 0 < C 1, C 2 < + ∞ such that the extinction probability starting with k individuals is bounded below by C 1 k -α 0 and above by C 2 k -α 1 for sufficiently large k. Moreover, a similar conclusion, which follows from a result of Höpfner, is presented along with some remarks.

Keywords

Population-size-dependent branching processbranching process in random environmentspopulation-size-dependent branching process in random environments

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60K37: Processes in random environments
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 1 , March 2004 , pp. 176 - 186
Copyright
Copyright © Applied Probability Trust 2004 

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.)

References

Cox, D. R., and Miller, H. D. (1965). The Theory of Stochastic Processes. Methuen, London.Google Scholar
Fang, D., and Wang, H. (1999). Population-size-dependent branching processes with independent stationary random environments. Chinese J. Appl. Prob. Statist. 15, 345–350.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Grey, D. R., and Lu, Z. (1991). Extinction probabilities of branching processes in random environments. In Selected Proc. Sheffield Symp. Appl. Prob. (IMS Lecture Notes Monogr. Ser. 18), Institute of Mathematical Statistics, Hayward, CA, pp. 205–211.Google Scholar
Grey, D. R., and Lu, Z. W. (1993). The asymptotic behaviour of extinction probability in the Smith–Wilkinson branching process. Adv. Appl. Prob. 25, 263–289.Google Scholar
Höpfner, R. (1985). On some classes of population-size-dependent branching processes. J. Appl. Prob. 22, 25–36.Google Scholar
Höpfner, R. (1985). A note on the probability of extinction in a class of population-size-dependent Galton–Watson processes. J. Appl. Prob. 22, 920–925.Google Scholar
Wang, H.-X. (1999). Extinction of population-size-dependent branching processes in random environments. J. Appl. Prob. 36, 146–154.Google Scholar