A Berry–Esseen bound for vector-valued martingales

This note provides a conditional Berry-Esseen bound for the sum of martingale difference sequence $\{X_i\}_{i=1}^n$ in $\mathbb{R}^d$, $d\ge 1$, adapted to filtration $\{\mathcal{F}_i\}_{i=1}^n$. We approximate distribution $S=\sum_{i=1}^n X_i$ given some $\sigma$-field $\mathcal{F}_0\subset \mathcal{F}_1$ by that mean-zero normal random vector having same variance $\mathcal{F}_0 $ as $S$. Assuming variances $\mathsf{E}[X_iX_i^{\top}\mid\mathcal{F}_{i-1}]$, $i\ge are $\mathcal{F}_0$-measurable and non-singular, third moments $\|X_i\|$, i\ge 1 $, $\mathcal{F}_0$ uniformly bounded, we present simple on Kolmogorov distance between $S$ its approximation which is order $O_{a.s.}([\ln(ed)]^{5/4}n^{-1/4})$.