In this paper we, under a very general dependence set-up,obtain a rate of convergence of the randomly stopped normalized log-lIkelI hood ratio statistic to standard normal variables. Two examples are taken Into account. Nonuniform rates of convergence and bounds for Lp norms are also obtained under some quite general conditions. Martingale techniques have been exploited throughout. © 1994, Taylor & Francis Group, LLC. All rights reserved.