A limit theorem for the maxima of the para-critical simple branching process

Abstract

LetMndenote the size of the largest amongst the firstngenerations of a simple branching process. It is shown for near critical processes with a finite offspring variance that the law ofMn/n, conditioned on no extinction at or beforen, has a non-defective weak limit. The proof uses a conditioned functional limit theorem deriving from the Feller-Lindvall (CB) diffusion limit for branching processes descended from increasing numbers of ancestors. Subsidiary results are given about hitting time laws for CB diffusions and Bessel processes.