A Berry-Esseen Bound for Vector-valued Martingales

TL;DR

This paper establishes a Berry-Esseen bound for the conditional distribution of sums of vector-valued martingale differences, providing a quantitative measure of how closely the sum approximates a normal distribution in high-dimensional settings.

Contribution

It introduces a new conditional Berry-Esseen bound for vector-valued martingales, extending classical results to a multivariate, conditional framework with explicit bounds.

Findings

Provides a bound of order O_{a.s.}([ ln(ed)]^{5/4} n^{-1/4}) on the Kolmogorov distance

Applicable to high-dimensional martingale difference sequences with bounded third moments

Assumes conditional variances are measurable and non-singular

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 $\mathsf{E}[X_iX_i^{\top}\mid\mathcal{F}_{i-1}]$, $i\ge 1$, are $\mathcal{F}_0$-measurable and non-singular, and the third conditional moments of $\|X_i\|$, $ 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 $O_{a.s.}([\ln(ed)]^{5/4}n^{-1/4})$.