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Genealogy for supercritical branching processes

Part of: Distribution theory - Probability Markov processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Andreas Nordvall Lagerås  and
Anders Martin-Löf
Andreas Nordvall Lagerås*
Affiliation:
Stockholm University
Anders Martin-Löf*
Affiliation:
Stockholm University
*
∗Postal address: Department of Mathematics, Stockholm University, Stockholm, SE-10691, Sweden.
∗Postal address: Department of Mathematics, Stockholm University, Stockholm, SE-10691, Sweden.
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Abstract

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We study the genealogy of so-called immortal branching processes, i.e. branching processes where each individual upon death is replaced by at least one new individual, and conclude that their marginal distributions are compound geometric. The result also implies that the limiting distributions of properly scaled supercritical branching processes are compound geometric. We exemplify our results with an expression for the marginal distribution for a class of branching processes that have recently appeared in the theory of coalescent processes and continuous stable random trees. The limiting distribution can be expressed in terms of the Fox H-function, and in special cases by the Meijer G-function.

Keywords

Supercritical branching processcompound geometric distributionFox H-functionMeijer G-function

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60E07: Infinitely divisible distributions; stable distributions 60F05: Central limit and other weak theorems 33C60: Hypergeometric integrals and functions defined by them ($E$, $G$, $H$ and $I$ functions)
Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 4 , December 2006 , pp. 1066 - 1076
Copyright
© Applied Probability Trust 2006 

References

Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.CrossRefGoogle Scholar
Berestycki, J., Berestycki, N. and Schweinsberg, J. (2006). Beta-coalescents and continuous stable random trees. To appear in Ann. Prob. Google Scholar
Bühler, W. J. (1968). Some results on the behaviour of branching processes. Theory Prob. Appl. 13, 52–64.CrossRefGoogle Scholar
Duquesne, T. and Le Gall, J.-F. (2002). Random trees, Lévy processes and spatial branching processes. Astérisque 281, 147pp.Google Scholar
Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.CrossRefGoogle Scholar
Mathai, A. M. and Saxena, R. K. (1978). The H-function with Applications in Statistics and Other Disciplines. John Wiley, New York.Google Scholar
Prudnikov, A. P., Brychkov, J. A. and Marichev, O. I. (1990). Integrals and Series, Vol. 3, More Special Functions. Gordon and Breach, New York.Google Scholar
Steutel, F. W. and van Harn, K. (2004). Infinite Divisibility of Probability Distributions on the Real Line. Marcel Dekker, New York.Google Scholar
Zolotarev, V. M. (1957). More exact statements of several theorems in the theory of branching processes. Theory Prob. Appl. 2, 245–253.CrossRefGoogle Scholar