A multivariate Berry--Esseen theorem with explicit constants

TL;DR

This paper establishes a multivariate Berry--Esseen theorem with explicit constants, providing bounds on normal approximation errors for sums of independent vectors, and improves Gaussian perimeter bounds for convex sets.

Contribution

It introduces a Lyapunov type bound with explicit constants in the multivariate CLT and enhances the Gaussian perimeter constant for convex sets.

Findings

Explicit error bounds for multivariate normal approximation

Improved constant for Gaussian perimeter of convex sets

Applicability to classes of measurable convex sets

Abstract

We provide a Lyapunov type bound in the multivariate central limit theorem for sums of independent, but not necessarily identically distributed random vectors. The error in the normal approximation is estimated for certain classes of sets, which include the class of measurable convex sets. The error bound is stated with explicit constants. The result is proved by means of Stein's method. In addition, we improve the constant in the bound of the Gaussian perimeter of convex sets.