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On the Maximal Offspring in a Critical Branching Process with Infinite Variance

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  14 July 2016

Jean Bertoin
Jean Bertoin*
Affiliation:
Université Pierre et Marie Curie
*
∗Postal address: Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris Cedex 05, France. Email address: jean.bertoin@upmc.fr
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Abstract

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We investigate the maximal number Mk of offspring amongst all individuals in a critical Galton-Watson process started with k ancestors. We show that when the reproduction law has a regularly varying tail with index -α for 1 < α < 2, then k-1Mk converges in distribution to a Frechet law with shape parameter 1 and scale parameter depending only on α.

Keywords

Branching processmaximal offspringFrechet distributionstable Lévy processextreme value theory

MSC classification

Secondary: 60F05: Central limit and other weak theorems 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 48 , Issue 2 , June 2011 , pp. 576 - 582
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
[2] Gradshteyn, I. S. and Ryzhik, I. M. (2007). Table of Integrals, Series, and Products, 7th edn. Elsevier/Academic, Amsterdam.Google Scholar
[3] Harris, T. E. (1952). First passage and recurrence distributions. Trans. Amer. Math. Soc. 73, 471–486.CrossRefGoogle Scholar
[4] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.CrossRefGoogle Scholar
[5] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
[6] Lebedev, A. V. (2008). Maxima of random properties of particles in supercritical branching processes. Moscow Univ. Math. Bull. 63, 175–178.CrossRefGoogle Scholar
[7] Lebedev, A. V. (2008). Maxima of random particles scores in Markov branching process with continuous time. Extremes 11, 203–216.CrossRefGoogle Scholar
[8] Pakes, A. G. (1998). Extreme order statistics on Galton–Watson trees. Metrika 47, 95–117.CrossRefGoogle Scholar
[9] Rahimov, I. and Yanev, G. P. (1999). On maximum family size in branching processes. J. Appl. Prob. 36, 632–643.CrossRefGoogle Scholar
[10] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.CrossRefGoogle Scholar
[11] Yanev, G. P. (2007). Revisiting offspring maxima in branching processes. Pliska Stud. Math. Bulgar. 18, 401–426.Google Scholar
[12] Yanev, G. P. (2008). A review of offspring extremes in branching processes. In Records and Branching Processes, eds Ahsanullah, M. and Yanev, G. P., Nova Science, pp. 127–145.Google Scholar