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Pré-Publication, Document De Travail Année : 2012

On largest offsprings in a critical branching process with finite variance

Jean Bertoin
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Résumé

We continue our study of the distribution of the maximal number $X^{\ast}_k$ of offsprings 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 $\bar F$ with index $-\alpha$ for $\alpha>2$ (and hence finite variance). We show that $X^{\ast}_k$ suitably normalized converges in distribution to a Frechet law with shape parameter $\alpha/2$; this contrasts sharply with the case $1<\alpha <2$ when the variance is infinite. More generally, we obtain a weak limit theorem for the offspring sequence ranked in the decreasing order, in terms of atoms of a certain doubly stochastic Poisson measure.
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hal-00733099 , version 1 (17-09-2012)

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Jean Bertoin. On largest offsprings in a critical branching process with finite variance. 2012. ⟨hal-00733099⟩
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