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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.)
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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References

[1] Agresti, A. (1974) Bounds on the extinction time distribution of a branching process. Adv. Appl. Prob. 6, 322–335.Google Scholar
[2] Agresti, A. (1975) On the extinction times of varying and random environment branching processes. J. Appl. Prob. 12, 39–46.Google Scholar
[3] Barnsley, M. F. and Elton, J. H. (1988) A new class of Markov processes for image encoding. Adv. Appl. Prob. 20, 14–32.Google Scholar
[4] Billingsley, P. (1965) Ergodic Theory and Information. Wiley, New York.Google Scholar
[5] Erdös, P. (1939) On a family of symmetric Bernoulli convolutions. Amer. J. Math. 61, 974–976.Google Scholar
[6] Erdös, P. (1940) On the smoothness properties of a family of Bernoulli convolutions. Amer. J. Math. 62, 180–186.Google Scholar
[7] Goldie, C. M. (1991) Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Prob. 1, 126–166.Google Scholar
[8] Grey, D. R. and Zhunwei, Lu (1993) The asymptotic behaviour of extinction probability in the Smith-Wilkinson branching process. Adv. Appl. Prob. 25, 263–289.Google Scholar
[9] Karlin, S. and Taylor, H. M. (1975) A First Course in Stochastic Processes, 2nd edn. Academic Press, New York.Google Scholar
[10] Keiding, N. and Nielsen, J. E. (1975) Branching processes with varying and random geometric offspring distributions. J. Appl. Prob. 12, 135–141.Google Scholar
[11] Zhunwei, Lu (1991) Survival of reproducing populations in random environments. Ph.D. Thesis, University of Sheffield.Google Scholar
[12] Smith, W. L. and Wilkinson, W. E. (1969) On branching processes in random environments. Ann. Math. Statist. 40, 814–827.Google Scholar
[13] Wilkinson, W. E. (1969) On calculating extinction probabilities for branching processes in random environments. J. Appl. Prob. 6, 478–492.Google Scholar