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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: luz1547@public.ty.sx.cn
∗∗ Postal address: Department of Mathematical Statistics, Chalmers University of Technology and Göteborg University, SE-412 96, Göteborg, Sweden. Email address: jagers@math.chalmers.se
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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 

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References

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