A high-dimensional CLT in ${\mathcal{W}}_2$ distance with near optimal convergence rate

Abstract

Let $X_1,\dots ,X_n$ be i.i.d. random vectors in ${\mathbb{R}}^d$ with $\Vert X_1\Vert \le \beta$. Then, we show that $\frac{1}{\sqrt{n}}(X_1+\dots +X_n)$ converges to a Gaussian in quadratic transportation (also known as "Kantorovich" or "Wasserstein") distance at a rate of $O\left(\frac{\sqrt{d}\beta \log n}{\sqrt{n}}\right)$, improving a result of Valiant and Valiant. The main feature of our theorem is that the rate of convergence is within $\log n$ of optimal for $n,d\to \infty$.

Updated introduction and various minor revisions, to appear in Probability Theory and Related Fields

References in corpus (2)

• Multivariate Normal Approximation by Stein's Method: The Concentration Inequality Approach
• An Exposition of Götze's Estimation of the Rate of Convergence in the Multivariate Central Limit Theorem