On largest offspring in a critical branching process with finite variance

Abstract

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.