On maximum family size in branching processes

Abstract

The number Y_n of offspring of the most prolific individual in the nth generation of a Bienaymé-Galton-Watson process is studied. The asymptotic behaviour of Y_n as n → ∞ may be viewed as an extreme value problem for i.i.d. random variables with random sample size. Limit theorems for both Y_n and EY_n provided that the offspring mean is finite are obtained using some convergence results for branching processes as well as a transfer limit lemma for maxima. Subcritical, critical and supercritical branching processes are considered separately.