On largest offspring in a critical branching process with finite variance. (English) Zbl 1277.60045

Bertoin, Jean

J. Appl. Probab. 50, No. 3, 791-800 (2013).

Summary: Continuing our work [J. Appl. Probab. 48, No. 2, 576–582 (2011; Zbl 1223.60020)], 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 \(\overline {F}\) with index \(-\alpha \) for \(\alpha >2\) (and, hence, finite variance). We show that \(X^{*}_{k}\) suitably normalized converges in distribution to a Fréchet law with shape parameter \(\alpha /2\); this contrasts sharply with the case \(1< \alpha <2\) for infinite variance. 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.

MSC:

60F05	Central limit and other weak theorems
60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60G57	Random measures

Keywords:

branching process; maximal offspring; extreme value theory; Cox process

Citations:

Zbl 1223.60020

References:

[1]	Athreya, K. B. (1988). On the maximum sequence in a critical branching process. Ann. Prob. 16 , 502-507. · Zbl 0643.60063
[2]	Bertoin, J. (2011). On the maximal offspring in a critical branching process with infinite variance. J. Appl. Prob. 48 , 576-582. · Zbl 1223.60020
[3]	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. · Zbl 0869.60077
[4]	Dwass, M. (1969). The total progeny in a branching process and a related random walk. J. Appl. Prob. 6 , 682-686. · Zbl 0192.54401
[5]	Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York. · Zbl 0996.60001
[6]	Lindvall, T. (1976). On the maximum of a branching process. Scand. J. Statist. 3 , 209-214. · Zbl 0344.60050
[7]	Pakes, A. G. (1998). Extreme order statistics on Galton-Watson trees. Metrika 47 , 95-117. · Zbl 1092.60505
[8]	Pitman, J. (2006). Combinatorial Stochastic Processes. (Lecture Notes Math. 1875 ), Springer, Berlin.
[9]	Rahimov, I. and Yanev, G. P. (1999). On maximum family size in branching processes. J. Appl. Prob. 36 , 632-643. · Zbl 0947.60085
[10]	Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York. · Zbl 0633.60001
[11]	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. · Zbl 1206.60078
[12]	Yanev, G. P. (2007). Revisiting offspring maxima in branching processes. Pliska Stud. Math. Bulgar. 18 , 401-426.
[13]	Yanev, G. P. (2008). A review of offspring extremes in branching processes. In Records and Branching Processes , eds M. Ahsanullah and G. P. Yanev, Nova Science, pp. 127-145.