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

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.


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


Zbl 1223.60020
Full Text: DOI arXiv Euclid


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