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.

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