## 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.

### MSC:

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

Zbl 1223.60020
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### References:

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