We investigate the maximal number Mk 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-1Mk converges in distribution to a Frechet law with shape parameter 1 and scale parameter depending only on α.
"On the maximal offspring in a critical branching process with infinite variance." J. Appl. Probab. 48 (2) 576 - 582, June 2011. https://doi.org/10.1239/jap/1308662646