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

We derive new Gaussian approximation 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 $n^{-1\mathrm{/}4}$ on the length of the sequence and of order $\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{l}\mathrm{o}\mathrm{g}(d)$ on the dimension. As an application, we derive a high-dimensional Berry-Esseen bound over hyper-rectangles for martingale sequences generated from Markov chains.