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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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References

Athreya, K. B. and Ney, P. E. (1972) Branching Process. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Biggins, J. D. and Bingham, N. H. (1991) Near-constancy phenomena in branching processes. Math. Proc. Camb. Phil. Soc. 110, 545–558.CrossRefGoogle Scholar
Biggins, J. D. and Nadarajah, S. (1993) Near-constancy of the Harris function in the simple branching process. Stoc. Models. 9. To appear.CrossRefGoogle Scholar
Biggins, J. D. and Shanbhag, D. N. (1981) Some divisibility problems in branching processes. Math. Proc. Camb. Phil. Soc. 90, 321–330.CrossRefGoogle Scholar
Bingham, N. H. (1988) On the limit of a supercritical branching process. J. Appl. Prob. 25A, 215–228.CrossRefGoogle Scholar
Boas, R. P. (1960) A Primer of Real Functions. Carus Mathematical Monographs 13, Mathematical Association of America, Washington, DC.Google Scholar
Dubuc, S. (1971) La densité de la loi-limite d'un processus en cascade expansif. Z. Wahrschleinlichkeitsth. 19, 281–290.CrossRefGoogle Scholar
Dubuc, S. (1982) Etude théorique et numérique de la fonction de Karlin-McGregor. J. Analyse Math. 42, 15–37.CrossRefGoogle Scholar
Ellis, R. S. (1984) Large deviations for a general class of random vectors. Ann. Prob. 12, 1–12.CrossRefGoogle Scholar
Flajolet, P. and Odlyzko, A. M. (1984) Limit distributions for coefficients of iterates of polynomials with applications to combinatorial enumeration. Math. Proc. Camb. Phil. Soc. 96, 237–273.CrossRefGoogle Scholar
Gärtner, J. (1977) On large deviations from the invariant measure. Theory Prob. Appl. 22, 24–39.CrossRefGoogle Scholar
Hardy, G. H., Littlewood, J. E. and Pólya, G. (1952) Inequalities, 2nd edn. Cambridge University Press.Google Scholar
Harris, T. E. (1948) Branching processes. Ann. Math. Statist. 41, 474–494.CrossRefGoogle Scholar
Hoppe, F. M. and Seneta, E. (1978) Analytical methods for discrete branching processes. In Branching Processes: Advances in Probability and Related Topics 5, ed. Joffe, A. and Ney, P., Marcel Dekker, New York, 219–261.Google Scholar
Kuczma, M., Choczewski, B. and Ger, R. (1990) Iterative Functional Equations. Cambridge University Press.CrossRefGoogle Scholar
Rockafellar, R. T. (1970) Convex Analysis. Princeton University Press.CrossRefGoogle Scholar