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Asymptotic behaviour of population-size-dependent branching processes in Markovian random environments

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Han-Xing Wang  and
Dafan Fang
Han-Xing Wang*
Affiliation:
Shanghai University
Dafan Fang*
Affiliation:
Yueyang Normal College
*
∗Postal address: Department of Mathematics, Shanghai University, Shanghai 201800, P.R. China.
∗∗Postal address: Department of Mathematics, Yueyang Normal College, Yueyang 414000, P.R. China.
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Abstract

A population-size-dependent branching process {Zn} is considered where the population's evolution is controlled by a Markovian environment process {ξn}. For this model, let mk,θ and be the mean and the variance respectively of the offspring distribution when the population size is k and a environment θ is given. Let B = {ω : Zn(ω) = 0 for some n} and q = P(B). The asymptotic behaviour of limnZn and is studied in the case where supθ|mk,θ − mθ| → 0 for some real numbers {mθ} such that infθmθ > 1. When the environmental sequence {ξn} is a irreducible positive recurrent Markov chain (particularly, when its state space is finite), certain extinction (q = 1) and non-certain extinction (q < 1) are studied.

Keywords

Markov chains in random environmentsbranching modelsextinction probabilities

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Short Communications
Information
Journal of Applied Probability , Volume 36 , Issue 2 , June 1999 , pp. 611 - 619
Copyright
Copyright © Applied Probability Trust 1999 

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References

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