Rates in the central limit theorem for random projections of martingales

In this paper, we consider partial sums of martingale differences weighted by random variables drawn uniformly on the sphere, and globally independent of the martingale differences. Combining Lindeberg's method and a series of arguments due to Bobkov, Chistyakov and G{ö}tze, we show that the Kolmogorov distance between the distribution of these weighted sums and the limiting Gaussian is "super-fast" of order (log n)^2 /n, under conditions allowing us to control the higher-order conditional moments of the martingale differences. We give an application of this result to the least squares estimator of the slope in the linear model with Gaussian design.