On the maximal offspring in a critical branching process with infinite variance

Abstract

We investigate the maximal number M_k of offspring amongst all individuals in a critical Galton-Watson process started with k ancestors. We show that when the reproduction law has a regularly varying tail with index -α for 1 < α < 2, then k^-1M_k converges in distribution to a Frechet law with shape parameter 1 and scale parameter depending only on α.