RATE OF CONVERGENCE IN A NON-LINEAR RENEWAL THEOREM

Abstract

Let x₁, x₂, ... be i. i. d. with normal mean θ and variance 1. For some function Λ(x), the non-linear process Z_n=nΛ(s_n/n), s_n=x₁+...+x_n, plays an important role in some sequential analyses. The stopping time T=inf{n≥1:Z_n≥a} and the asymptotic distribution P_θ{Z_T-a≤x} of Z_T-a have been considered by Lai and Siegmund. This paper is concerned with the rate of convergence, which is shown to be of the order 0 (a^-1log⁵a) as a→∞.