Hostname: page-component-77c89778f8-fv566 Total loading time: 0 Render date: 2024-07-18T18:52:55.899Z Has data issue: false hasContentIssue false
  • English
  • Français

Extinction of population-size-dependent branching processes in random environments

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Han-xing Wang
Han-xing Wang*
Affiliation:
Shanghai University
*
∗Postal address: Department of Mathematics, Shanghai University, Shanghai, 201800, P.R. China. Email address: susts@guomai.sh.cn.
Get access Rights & Permissions [Opens in a new window]

Abstract

We generalize a population-size-dependent branching process to a more general branching model called the population-size-dependent branching process in random environments. For the model where {Zn}n≥0 is associated with the stationary environment ξ− = {ξn}n≥0, let B = {ω : Zn(ω) = for some n}, and q(ξ−) = P(B | ξ−, Z0 = 1). The result is that P(q(̅ξ) = 1) is either 1 or 0, and sufficient conditions for certain extinction (i.e. P(q(ξ−) = 1) = 1) and for non-certain extinction (i.e. P(q(ξ−) < 1) = 1) are obtained for the model.

Keywords

Stochastic population modelsMarkov chains in random environmentsextinction probabilities

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 1 , March 1999 , pp. 146 - 154
Copyright
Copyright © Applied Probability Trust 1999 

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

Athreya, K. B., and Karlin, S. (1971). On branching processes with random environments: I extinction probabilities. Ann. Math. Statist. 42, 1499–1520.CrossRefGoogle Scholar
Cogburn, R. (1984). The ergodic theory of Markov chains in random environments. Z.Wahrscheinlichkeitsth 66, 109–128.Google Scholar
Klebaner, F. C. (1984). On population-size-dependent branching processes. Adv. Appl. Prob. 16, 30–55.Google Scholar
Pierre Loti Viaud, D. (1994). A strong law and a central limit theorem for controlled Galton–Watson processes. J. Appl. Prob. 31, 22–37.Google Scholar
Smith, W. L., and Wilkinson, W. E. (1969). On branching processes in random environments. Ann. Math. Statist. 40, 814–827.Google Scholar