On the rates of convergence in limit theorems for one-sided stopping rules

Let X1, X2, ... be i.i.d. r.vs. with EX=µ>0 and E(X-µ)2 = σ2<σ. Let Sk = X 1 + ... + Xk and Tr = Inf {k ⩾ 1|Sk > rk± }, r > 0, ± > 0. Such stopping rules are extensively used by Darling & Robbins (1967), Robbins (1970), Robbins & Siegmund (1970), and others. Define Zr = (Tr −λ(r))/(σµ−1(l−±)-1√γ(r)) where γ(x)=(x/µ)1/², x ⩾0, ²=1-±, and 0⩽±<1. Kennedy obtained the rate of convergence of Zr , to the standard normal distribution when ±=0. We have extended his result to the case ±>0.