Functional equations and the Galton-Watson process

Abstract

In the present exposition we are concerned only with the simple Galton-Watson process, initiated by a single ancestor (Harris (1963), Chapter I). Letdenote the probability generating function of the offspring distribution of a single individual. Our fundamental assumption, which holds throughout the sequel, is thatfj≠ 1,j= 0,1,2, …; in particular circumstances it shall be necessary to strengthen this to 0 <f0≡F(0) < 1, which is the relevant assumption when extinction behaviour is to be considered. (Even so, our assumptions will always differ slightly from those of Harris (1963), p. 5.)