Domains of attraction for the subcritical Galton-Watson branching process

Abstract

Let F(z) = σ fjzj be the generating function for the offspring distribution {fj } from a single ancestor in the usual Galton-Watson process. It is well-known (see Harris [1]) that if Π(z) is the generating function of the distribution of ancestors in the 0th generation, the distribution of offspring at the nth generation has generating function where F n (z), the nth functional iterate of F(z), gives the distribution of offspring at the nth generation from a single ancestor.