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On constant tail behaviour for the limiting random variable in a supercritical branching process

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

B. M. Hambly
B. M. Hambly*
Affiliation:
University of Edinburgh
*
∗Postal address: Department of Mathematics and Statistics, University of Edinburgh, King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, UK.
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Abstract

We examine a family of supercritical branching processes and compute the density of the limiting random variable, W, for their normalized population size. In this example the left tail of W decays exponentially and there is no oscillation in this tail as typically observed. The branching process is embedded in the n-adic rational random walk approximation to Brownian motion on [0, 1]. This connection allows the explicit computation of the density of W.

Keywords

BROWNIAN MOTIONSIERPINSKI GASKET

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J65: Brownian motion
Type
Short Communications
Information
Journal of Applied Probability , Volume 32 , Issue 1 , March 1995 , pp. 267 - 273
Copyright
Copyright © Applied Probability Trust 1995 

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References

[1] Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer-Verlag, Berlin.CrossRefGoogle Scholar
[2] Barlow, M. T. and Perkins, E. A. (1988) Brownian motion on the Sierpinski gasket. Prob. Theory Rel. Fields. 79, 543–624.CrossRefGoogle Scholar
[3] Biggins, J. D. and Bingham, N. H. (1993) Large deviations for the supercritical branching process. Adv. Appl. Prob. 25, 757–772.CrossRefGoogle Scholar
[4] Dubuc, S. (1982) Etude théorique et numérique de la fonction de Karlin-McGregor. J. Anal. Math. 42, 15–37.CrossRefGoogle Scholar
[5] Gradshteyn, I. S. and Ryzhik, I. M. (1980) Tables of Integrals, Series and Products. Academic Press, San Diego.Google Scholar
[6] Harris, T. E. (1948) Branching processes. Ann. Math. Statist. 19, 474–494.CrossRefGoogle Scholar
[7] Karatzas, I. and Shreve, S. E. (1988) Brownian Motion and Stochastic Calculus. Springer-Verlag, New York.CrossRefGoogle Scholar