A Berry-Esseen Bound for Vector-valued Martingales

Abstract

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