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

TL;DR

This paper establishes new Gaussian approximation bounds for multivariate martingale difference sequences in the Kolmogorov distance, highlighting their dependence on sequence length and dimension.

Contribution

It introduces novel Berry-Esseen bounds for high-dimensional martingale sequences with explicit dependence on sequence length and dimension.

Findings

Bounds depend on sequence length as n^{-1/4}

Dimension dependence is polylogarithmic

Application to high-dimensional Markov chain martingales

Abstract

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 $n^{-1/4}$ 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.