Exact convergence rates in the central limit theorem for a class of martingales

We give optimal convergence rates in the central limit theorem for a large class of martingale difference sequences with bounded third moments. The rates depend on the behaviour of the conditional variances and for stationary sequences the rate $n^{-1\mathrm{/}2}\log n$ is reached. We give interesting examples of martingales with unbounded increments which belong to the considered class.