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

Abstract

Given an isotropic probability measure $\mu$ on ${\mathbb{R}}^d$ with $\mathrm{d}\mu \left(x\right)={\left(\varrho \left(x\right)\right)}^{-\alpha }\mathrm{d}x$, where $\alpha >d+1$ and $\varrho :{\mathbb{R}}^d\to \left(0,+\infty \right)$ is a continuous function and uniformly convex (${\nabla }^2\varrho \ge {\epsilon }_0\mathrm{Id}$). 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 $c_{{\epsilon }_0}\sqrt{\frac{d}{n}}$. The general case (i.e., $\varrho$ is only convex and continuous) remains open.

This is not a research paper, but an internal draft containing mistakes which should have not be made publicly available