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Large deviations in the supercritical branching process

Published online by Cambridge University Press:  01 July 2016

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.

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.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 1993 

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