On Berry Esseen type estimates for randomized Martingales in the non stationary setting

In this paper, we consider partial sums of triangular martingale differences weighted by random variables drawn uniformly on the sphere, and globally independent of the martingale differences. Starting from the so-called principle of conditioning and using some arguments developed by Klartag-Sodin and Bobkov-Chistyakov-G{ö}tze, we give some upper bounds for the Kolmogorov distance between the distribution of these weighted sums and a Normal distribution. Under some conditions on the conditional variances of the martingale differences, the obtained rates are always faster than those obtained in case of usual partial sums.