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 {Xi}i=1n in Rd, d≥1, adapted to a filtration {Fi}i=1n. We approximate the conditional distribution of S=∑i=1nXi given a sub-σ-field F0⊂F1 by that of a mean zero normal random vector having the same conditional variance given F0 as the vector S. Assuming that the conditional variances E[XiXi⊤∣Fi−1], i≥1, are F0-measurable and non-singular, and the third conditional moments of ‖Xi‖, i≥1, given F0 are uniformly bounded, we present a simple bound on the conditional Kolmogorov distance between S and its approximation given F0 which is of order Oa.s.([ln(ed)]5/4n−1/4).