论文标题
对数孔分布的尖锐的高维中心极限定理
Sharp High-dimensional Central Limit Theorems for Log-concave Distributions
论文作者
论文摘要
令$ x_1，\ dots，x_n $ be i.i.d. $ \ mathbb r^d $中的log-conconcave随机向量，均值0和协方差矩阵$σ$。我们研究了$ w = n^{ - 1/2} \ sum_ {i = 1}^nx_i $的正常近似错误的问题，并且明确依赖于尺寸$ d $。具体来说，我们在$σ$上没有任何限制，我们表明$ \ Mathbb r^d $中矩形的近似错误由$ c（\ log^{13}（dn）/n）/n）^{1/2} $界定，对于某些通用常数$ c $。此外，如果Kannan-Lovász-Simonovits（KLS）频谱差距猜想是正确的，则该界限可以改进到$ c（\ log^{3}（dn）/n）/n）/n）^{1/2} $。在$ n $和$ d $的制度$ \ log n = o（\ log d）$中，这种改进的界限都是最佳的。我们还提供了所有$ p \ geq2 $的$ p $ - wasserstein界限，并且对于此正常近似错误，Cramér类型中度偏差结果，并且在KLS猜想下都是最佳的。为了证明这些界限，我们开发了一种新的高斯耦合不平等，该耦合不等式几乎可以为每$ p \ geq2 $ $ p $ wasserstein距离的预计版本提供几乎无尺寸的范围。我们通过结合Stein的方法和Eldan的随机定位程序来证明这种耦合不平等。
Let $X_1,\dots,X_n$ be i.i.d. log-concave random vectors in $\mathbb R^d$ with mean 0 and covariance matrix $Σ$. 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 $Σ$, 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-Lovász-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 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 $p\geq2$. We prove this coupling inequality by combining Stein's method and Eldan's stochastic localization procedure.
