The CLT in high dimensions: quantitative bounds via martingale embedding

TL;DR

This paper presents a new martingale embedding method to derive quantitative convergence rates for the high-dimensional CLT, improving bounds in transportation distance and entropy for various classes of random vectors.

Contribution

The paper introduces a novel martingale embedding approach that yields the first non-asymptotic entropic CLT rate in arbitrary dimensions and improves existing bounds under log-concavity assumptions.

Findings

Improved quadratic Wasserstein convergence bounds for bounded vectors

First non-asymptotic entropic CLT rate in arbitrary dimensions

Enhanced convergence bounds under log-concavity and strong log-concavity

Abstract

We introduce a new method for obtaining quantitative convergence rates for the central limit theorem (CLT) in a high dimensional setting. Using our method, we obtain several new bounds for convergence in transportation distance and entropy, and in particular: (a) We improve the best known bound, obtained by the third named author, for convergence in quadratic Wasserstein transportation distance for bounded random vectors; (b) We derive the first non-asymptotic convergence rate for the entropic CLT in arbitrary dimension, for general log-concave random vectors; (c) We give an improved bound for convergence in transportation distance under a log-concavity assumption and improvements for both metrics under the assumption of strong log-concavity. Our method is based on martingale embeddings and specifically on the Skorokhod embedding constructed by the first named author.