Rates of convergence in the distances of Kolmogorov and Wasserstein for standardized martingales

We give some rates of convergence in the distances of Kolmogorov and Wasserstein for standardized martingales with differences having finite variances. For the Kolmogorov distances, we present some exact Berry-Esseen bounds for martingales, which generalizes some Berry-Esseen bounds due to Bolthausen. In consequence, a Lindeberg type condition for the martingale central limit theorem is obtained. For the Wasserstein distance, with Stein's method and Lindeberg's telescoping sum argument, the rates of convergence in martingale central limit theorems recover the classical rates for sums of i.i.d. random variables, and therefore they are believed to be optimal.