The Galton-Watson process with infinite mean

Abstract

Let Zn be the number of descendents in the nth generation of a simple Galton-Watson branching process, initiated by a single progenitor, Z0 = 1. If E(Z1) < ∞ the limiting distribution of Zn is known in some detail, and a comprehensive account is given in Seneta [1]. If E{Z1) = ∞ (the “explosive” case) the behavior of the distribution of Zn for large n seems not to be known. As shown by Seneta [1], there are no constants cn such that cnZn has a non-degenerate limiting distribution but it turns out that, under conditions given below, log(Zn + 1) has a limiting distribution.