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On the Survival Probability of a Branching Process in a Random Environment

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

J. C. D'souza  and
B. M. Hambly
J. C. D'souza*
Affiliation:
University of Aberdeen
B. M. Hambly*
Affiliation:
University of Edinburgh
*
∗ Postal address: Department of Mathematical Sciences, University of Aberdeen, Edward Wright Building, Dunbar Street, Aberdeen, AB24 3QY, UK.
∗∗ Postal address: Department of Mathematics and Statistics, University of Edinburgh, The King's Buildings, Mayfield Road, Edinburgh, EH9 3JZ, UK.
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Abstract

We derive a variety of estimates for the survival probability of a branching process in a random environment. There are three cases of interest, the critical, weakly and strongly subcritical. The large deviation result, first obtained by Dekking for the class of finite state space i.i.d. environments, is shown to hold in more general environments. We also obtain some finer convergence results.

Keywords

BRANCHING PROCESSESRANDOM ENVIRONMENTSVARYING ENVIRONMENTSGALTON-WATSON PROCESSSURVIVAL PROBABILITYRANDOM WALKSEXTINCTION TIMELARGE DEVIATIONS

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 29 , Issue 1 , March 1997 , pp. 38 - 55
Copyright
Copyright © Applied Probability Trust 1997 

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