A rate of convergence result for the super-critical Galton-Watson process

Abstract

Let Z0 = 1, Z1, Z2, ··· denote a super-critical Galton-Watson process whose non-degenerate offspring distribution has probability generating function where 1 < m = EZ 1 < ∞. The Galton-Watson process evolves in such a way that the generating function Fn (s) of Zn is the nth functional iterate of F(s). The convergence problem for Z n, when appropriately normed, has been studied by quite a number of authors; for an ultimate form see Heyde [2]. However, no information has previously been obtained on the rate of such convergence. We shall here suppose that in which case Wn = m –nZn converges almost surely to a non-degenerate random variable W as n → ∞ (Harris [1], p. 13). It is our object to establish the following result on the rate of convergence of Wn to W.