On largest offsprings in a critical branching process with finite variance

Abstract

We continue our study of the distribution of the maximal number $X_k^{\ast }$ 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 $X_k^{\ast }$ 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.