A Rate of Convergence Result for the Super-Critical Galton-Watson Process

Abstract

Let Z 0 = 1, Z 1, Z 2,... denote a super-critical Galton–Watson process whose non-degenerate offspring distribution has probability generating function \(F\left(s\right)=\sum\nolimits^{\infty}_{{j}=0}s^j \mathbf{P}{\rm r}\left(\mathbf{Z}_1=j\right), 0\leqq s \leqq 1,\) where 1 < m = E Z 1 < ∞. The Galton–Watson process evolves in such a way that the generating function F n(S) of Z n 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 suppose that E Z 1 2 < ∞ in which case W n = m -n Z n 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 W n ot W.