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The fractional linear probability generating function in the random environment branching process

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

D. R. Grey  and
Lu Zhunwei
D. R. Grey*
Affiliation:
University of Sheffield
Lu Zhunwei*
Affiliation:
University of Sheffield
*
∗ Postal address: School of Mathematics and Statistics, The University of Sheffield, PO Box 597, Sheffield S10 2UN, UK.
∗∗ Current address: Department of Mathematics, Physics and Mechanics, Taiyuan University of Technology, Taiyuan, Shanxi Province, The People's Republic of China.
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Abstract

In a branching process with random environments, the probability of ultimate extinction is a function of the environment sequence, and is therefore a random variable. Explicit results about the distribution of this random variable are difficult to obtain in general. Here we assume independent and identically distributed environments and use the special properties of fractional linear generating functions to derive some explicit distributions, which may be singular or absolutely continuous, depending on the values of certain parameters. We also consider briefly tail behaviour close to 1, and provide an extension to cases where probability generating functions are not fractional linear.

Keywords

SMITH-WILKINSON BRANCHING PROCESSDUAL PROCESSEXTINCTION PROBABILITY

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 31 , Issue 1 , March 1994 , pp. 38 - 47
Copyright
Copyright © Applied Probability Trust 1994 

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