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The Rates of Growth of the Galton–Watson Process in Varying Environments

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

J. C. D'Souza
J. C. D'Souza*
Affiliation:
Heriot-Watt University
*
* Present address: Department of Mathematical Sciences, The University, Aberdeen AB9 2TY, UK.
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Abstract

Let {Zn} be a supercritical Galton–Watson process in varying environments, and W be the limit of the non-negative martingale {Zn/EZn}. Under a condition which ensures that W is not identically equal to zero we give an upper bound on the possible rates of growth of the process on the set {W = 0}, and find a sufficient condition for the process to have only one rate of growth. We also give an example of a process whose offspring distributions have bounded pth moments, for some p > 1, and which has an infinite number of rates of growth.

Keywords

BRANCHING PROCESSESMARTINGALESOFFSPRING DISTRIBUTIONS

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 26 , Issue 3 , September 1994 , pp. 698 - 714
Copyright
Copyright © Applied Probability Trust 1994 

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