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Large deviations in the supercritical branching process

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  01 July 2016

J. D. Biggins  and
N. H. Bingham
J. D. Biggins*
Affiliation:
University of Sheffield
N. H. Bingham*
Affiliation:
Royal Holloway and Bedford New College
*
* Postal address: School of Mathematics and Statistics, The University of Sheffield, PO Box 597, Sheffield S10 2UN, UK.
** Postal address: Mathematics Department, Royal Holloway and Bedford New College, Egham Hill, Egham, Surrey TW20 0EX, UK.
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Abstract

The tail behaviour of the limit of the normalized population size in the simple supercritical branching process, W, is studied. Most of the results concern those cases when a tail of the distribution function of W decays exponentially quickly. In essence, knowledge of the behaviour of transforms can be combined with some ‘large-deviation' theory to get detailed information on the oscillation of the distribution function of W near zero or at infinity. In particular we show how an old result of Harris (1948) on the asymptotics of the moment-generating function of W translates to tail behaviour.

Keywords

MULTIPLICATIVELY PERIODICTAIL BEHAVIOUR

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F10: Large deviations
Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 4 , December 1993 , pp. 757 - 772
Copyright
Copyright © Applied Probability Trust 1993 

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