Berry-Esseen bounds for multivariate martingale difference sequences in the Kolmogorov distance

Abstract

We derive new Gaussian approximations for finite martingale difference sequences in ${\mathbb{R}}^d$ with respect to the Kolmogorov distance. Under appropriate conditions, our bounds exhibit a dependence of order $n^{-1/4}$ on the length of the sequence and of order $\mathrm{polylog}(d)$ on the dimension. As an application, we derive a high-dimensional Berry-Esseen bound over hyper-rectangles for martingale sequences generated from Markov chains.