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

Let $X_1,\dots ,X_n$ be i.i.d. log-concave random vectors in ${\mathbb{R}}^d$ with mean 0 and covariance matrix $\mathrm{\Sigma }$. We study the problem of quantifying the normal approximation error for $W=n^{-1\mathrm{/}2}{\sum }_{i=1}^n{X}_i$ with explicit dependence on the dimension $d$. Specifically, without any restriction on $\mathrm{\Sigma }$, we show that the approximation error over rectangles in ${\mathbb{R}}^d$ is bounded by $C({\log }^{13}(dn)\mathrm{/}n{)}^{1\mathrm{/}2}$ for some universal constant $C$. Moreover, if the Kannan-Lovász-Simonovits (KLS) spectral gap conjecture is true, this bound can be improved to $C({\log }^3(dn)\mathrm{/}n{)}^{1\mathrm{/}2}$. 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 $p\ge 2$ and a Cramér 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 $p\ge 2$. We prove this coupling inequality by combining Stein’s method and Eldan’s stochastic localization procedure.