Rates of Convergence in the Martingale Central Limit Theorem

Abstract

We obtain a nonuniform estimate of the rate of convergence in the martingale central limit theorem for convergence to mixtures of normal distributions. The uniform rates of convergence obtained by several other authors are special cases of our nonuniform estimate. We also obtain a rate of convergence in B. M. Brown's central limit theorem, assuming only Brown's elementary conditions. This result is a martingale analogue of Feller's generalization of the Berry-Esseen theorem.