The Critical Galton-Watson Process Without Further Power Moments

In this paper we prove a conditional limit theorem for a critical Galton-Watson branching process {Z_n ; n ≥ 0} with offspring generating function s + (1 − s)L((1 − s)^−1), where L(x) is slowly varying. In contrast to a well-known theorem of Slack (1968), (1972) we use a functional normalization, which gives an exponential limit. We also give an alternative proof of Sze's (1976) result on the asymptotic behavior of the nonextinction probability