We investigate the maximal number Mk of offspringamongst all individuals in a critical Galton-Watson process started withk ancestors. We show that when the reproduction law has a regularlyvarying tail with index -α for 1 < α < 2, thenk-1Mk converges in distribution to aFrechet law with shape parameter 1 and scale parameter depending only onα.
"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