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On Largest Offspring in a Critical Branching Process with Finite Variance

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  30 January 2018

Jean Bertoin
Jean Bertoin*
Affiliation:
Universität Zürich
*
∗ Postal address: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland. Email address: jean.bertoin@math.uzh.ch
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Abstract

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Continuing the work in Bertoin (2011) we study the distribution of the maximal number X*k of offspring amongst all individuals in a critical Galton‒Watson process started with k ancestors, treating the case when the reproduction law has a regularly varying tail F̅ with index −α for α > 2 (and, hence, finite variance). We show that X*k suitably normalized converges in distribution to a Fréchet law with shape parameter α/2; this contrasts sharply with the case 1< α<2 when the variance is infinite. More generally, we obtain a weak limit theorem for the offspring sequence ranked in decreasing order, in terms of atoms of a certain doubly stochastic Poisson measure.

Keywords

Branching processmaximal offspringextreme value theoryCox process

MSC classification

Primary: 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 50 , Issue 3 , September 2013 , pp. 791 - 800
Copyright
© Applied Probability Trust 

References

Athreya, K. B. (1988). On the maximum sequence in a critical branching process. Ann. Prob. 16, 502–507.CrossRefGoogle Scholar
Bertoin, J. (2011). On the maximal offspring in a critical branching process with infinite variance. J. Appl. Prob. 48, 576–582.CrossRefGoogle Scholar
Borovkov, K. A. and Vatutin, V. A. (1996). On distribution tails and expectations of maxima in critical branching processes. J. Appl. Prob. 33, 614–622.CrossRefGoogle Scholar
Dwass, M. (1969). The total progeny in a branching process and a related random walk. J. Appl. Prob. 6, 682–686.Google Scholar
Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.Google Scholar
Lindvall, T. (1976). On the maximum of a branching process. Scand. J. Statist. 3, 209–214.Google Scholar
Pakes, A. G. (1998). Extreme order statistics on Galton-Watson trees. Metrika 47, 95–117.CrossRefGoogle Scholar
Pitman, J. (2006). Combinatorial Stochastic Processes. (Lecture Notes Math. 1875), Springer, Berlin.Google Scholar
Rahimov, I. and Yanev, G. P. (1999). On maximum family size in branching processes. J. Appl. Prob. 36, 632–643.Google Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.CrossRefGoogle Scholar
Vatutin, V. A., Wachtel, V. and Fleischmann, K. (2008). Critical Galton-Watson branching processes: the maximum of the total number of particles within a large window. Theory Prob. Appl. 52, 470–492.CrossRefGoogle Scholar
Yanev, G. P. (2007). Revisiting offspring maxima in branching processes. Pliska Stud. Math. Bulgar. 18, 401–426.Google Scholar
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