Covariance Representations, $L^p$-Poincaré Inequalities, Stein's Kernels and High Dimensional CLTs

Abstract

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 ${\mathbb{R}}^d$, $d\ge 1$. 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 ($p\ge 2$) for the standard Gaussian probability measure on ${\mathbb{R}}^d$ based on a covariance representation. Moreover, for the nondegenerate symmetric $\hat{I}\pm$-stable case, $\hat{I}\pm \in (1,2)$, we obtain $L^p$-Poincaré and pseudo-Poincaré inequalities, for $p\in (1,\hat{I}\pm )$, 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.

58 pages