A lemma on the Galton-Watson process and some of its consequences

Abstract

Many of the known limit theorems on the Galton-Watson process are immediate corollaries of the elementary lemma given below. This lemma enables us to calculate easily limits of various types of conditional distributions, special cases of which had earlier been tackled by diverse and more complicated methods. Let F(x) = j opljxj be the distribution generating function of the offspring of one individual in a Galton-Watson process. We assume 0 0. We set Fo(x)=x, Fn+l(x) =F(Fn(x)) and denote by q the probability of extinction of the progeny of one individual. Then q =1 or 1. Further Fn(x) T q if 0? x<q, Fn(x) I q if q<x<1 while