A Berry-Esseen type inequality for convex bodies with an unconditional basis

TL;DR

This paper establishes a precise rate of convergence in the central limit theorem for high-dimensional random vectors with unconditional, log-concave densities, using advanced analysis of convex domains and measure transportation.

Contribution

It introduces a sharp Berry-Esseen type inequality specifically for convex bodies with an unconditional basis, advancing the understanding of convergence rates in high-dimensional probability.

Findings

Provides a sharp convergence rate in the CLT for unconditional, log-concave vectors.

Utilizes analysis of the Neumann Laplacian on convex domains.

Employs optimal transportation theory to derive results.

Abstract

We provide a sharp rate of convergence in the central limit theorem for random vectors with an unconditional, log-concave density. The argument relies on analysis of the Neumann laplacian on convex domains and on the theory of optimal transportation of measures.