A Berry-Esseen bound for vector-valued martingales

This note provides a conditional Berry-Esseen bound for the sum of a martingale difference sequence {X-i}(i=1)(n) in d >= 1 the adapted to a filtration {F-i}(i=1)(n.) We approximate the conditional distribution of S = Sigma(n)(i=1) X-i given a sub-sigma-field F-0 subset of F-1 by that of a mean zero normal random vector having the same conditional variance given F-0 as the vector S. Assuming that the conditional variances E[XiXiT | Fi-1], i >= 1, are F-0-measurable and non-singular, and the third conditional moments of ||X-i||i >= 1, given F-0 are uniformly bounded, we present a simple bound on the conditional Kolmogorov distance between S and its approximation given F-0 which is of order O-a.s.([ln(ed)](5/4)n(-1/4)).(c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).