论文标题
协方差表示，$ l^p $-poincaré不平等，Stein的内核和高维clts
Covariance Representations, $L^p$-Poincaré Inequalities, Stein's Kernels and High Dimensional CLTs
论文作者
论文摘要
我们探索协方差表示，bismut型公式和Stein方法之间的联系。首先，使用封闭对称形式的理论，我们得出了$ \ mathbb {r}^d $，$ d \ geq 1 $的几种知名概率度量的协方差表示。当有强大的梯度范围时，这些协方差表示立即导致$ l^p $ - $ l^q $协方差估算，所有$ p \ in（1， +\ infty）$和$ q = p/（p/（p-1）$。然后，我们根据协方差表示，对标准高斯概率措施进行了众所周知的$ l^p $-poincaré不平等（$ p \ geq 2 $），以$ \ mathbb {r}^d $上的标准概率度量。此外，对于非成熟的对称$α$稳定案例，$α\ in（1,2）$，我们获得了$ l^p $-poincaré和pseudo-poincaré不平等，以$ p \ in（1，α）$的$ p \ in（1，α）$，通过对各种bismut-type type typeposals的详细分析。最后，使用封闭形式技术对Stein的内核的构建，当限制高斯概率度量是各向异性时，我们以$ 1 $ - 沃瑟氏级距离获得定量的高维CLT。对参数的依赖性完全是显式的，并且收敛速率很清晰。
We explore connections between covariance representations, Bismut-type formulas and Stein's method. First, using the theory of closed symmetric forms, we derive covariance representations for several well-known probability measures on $\mathbb{R}^d$, $d \geq 1$. When strong gradient bounds are available, these covariance representations immediately lead to $L^p$-$L^q$ covariance estimates, for all $p \in (1, +\infty)$ and $q = p/(p-1)$. Then, we revisit the well-known $L^p$-Poincaré inequalities ($p \geq 2$) for the standard Gaussian probability measure on $\mathbb{R}^d$ based on a covariance representation. Moreover, for the nondegenerate symmetric $α$-stable case, $α\in (1,2)$, we obtain $L^p$-Poincaré and pseudo-Poincaré inequalities, for $p \in (1, α)$, via a detailed analysis of the various Bismut-type formulas at our disposal. Finally, using the construction of Stein's kernels by closed forms techniques, we obtain quantitative high-dimensional CLTs in $1$-Wasserstein distance when the limiting Gaussian probability measure is anisotropic. The dependence on the parameters is completely explicit and the rates of convergence are sharp.
