The maximum in critical Galton–Watson and birth and death processes

Abstract

In this paper recent results by Weiner [10] on Mn:= max{Z0, · ··, Zn} are strengthened and generalized, where (Zn)n is a critical Galton–Watson branching process with finite and positive offspring variance and Z0 ≡ 1. It is shown that Explicit asymptotic bounds are given for with . If (Zn)n has a linear fractional offspring distribution, it can be embedded in a critical birth and death process (Ẑ t)t. Using martingale methods one obtains thereof.These results generalize to the case Z0 ≡ k.