Stein kernels for $q$-moment measures and new bounds for the rate of convergence in the central limit theorem

TL;DR

This paper introduces Stein kernels for $q$-moment measures to derive new bounds on the convergence rate in the central limit theorem for specific isotropic probability measures with convex densities.

Contribution

It develops Stein kernel techniques for $q$-moment measures to establish explicit convergence rates in the CLT for measures with convex densities.

Findings

Convergence rate of order $rac{1}{ oot n}$ for certain measures.

Explicit bounds depending on convexity parameters.

Extension of Stein kernel methods to $q$-moment measures.

Abstract

Given an isotropic probability measure $\mu$ on ${\mathbb R}^d$ with ${\rm d}\mu \left( x \right) = {\left( {\varrho \left( x \right)} \right)^{ - \alpha }}{\rm d}x$, where $\alpha > d + 1$ and $\varrho :{{\mathbb R}^d} \to \left( {0, + \infty } \right)$ is a continuous function and uniformly convex (${\nabla ^2}\varrho \ge {\varepsilon_0}{\rm {Id}}$). By using Stein kernels for $\left( {\alpha - d} \right)$-moment measures, we prove that the rates of convergence in the central limit theorem with sequence of i.i.d. random variables ${X_1},{X_2},...,{X_n}$ of the law $\mu$, to be of form $c_{{\varepsilon}_0}\,\sqrt {\dfrac{d}{n}} $. The general case (i.e., $\varrho$ is only convex and continuous) remains open.