Sharp High-dimensional Central Limit Theorems for Log-concave Distributions

Abstract

Let $X_1,\dots,X_n$ be i.i.d. log-concave random vectors in $\mathbb R^d$ with mean 0 and covariance matrix $\Sigma$. We study the problem of quantifying the normal approximation error for $W=n^{-1/2}\sum_{i=1}^nX_i$ with explicit dependence on the dimension $d$. Specifically, without any restriction on $\Sigma$, we show that the approximation error over rectangles in $\mathbb R^d$ is bounded by $C(\log^{13}(dn)/n)^{1/2}$ for some universal constant $C$. Moreover, if the Kannan-Lov\'asz-Simonovits (KLS) spectral gap conjecture is true, this bound can be improved to $C(\log^{3}(dn)/n)^{1/2}$. This improved bound is optimal in terms of both $n$ and $d$ in the regime $\log n=O(\log d)$. We also give $p$-Wasserstein bounds with all $p\geq2$ and a Cram\'er type moderate deviation result for this normal approximation error, and they are all optimal under the KLS conjecture. To prove these bounds, we develop a new Gaussian coupling inequality that gives almost dimension-free bounds for projected versions of $p$-Wasserstein distance for every $p\geq2$. We prove this coupling inequality by combining Stein's method and Eldan's stochastic localization procedure.