On the Rate of Convergence in the Central Limit Theorem for Martingales with Discrete and Continuous Time

Abstract

Heyde and Brown (1970) established a bound on the rate of convergence in the central limit theorem for discrete time martingales having finite moments of order $2 + 2\delta$ with $0 < \delta \leq 1$. In the present paper a modification of the methods developed by Bolthausen (1982) is applied to show the validity of this result for all $\delta > 0$. Moreover, an example is constructed demonstrating that this bound is asymptotically exact for all $\delta > 0$. The result for discrete time martingales is then used to derive the corresponding bound on the rate of convergence in the central limit theorem for locally square integrable martingales with continuous time.