# Higher-order Stein kernels for Gaussian approximation

## TL;DR

This paper introduces higher-order Stein kernels for Gaussian approximation, extending classical kernels with higher derivatives, leading to improved convergence rates in the multidimensional CLT under certain conditions.

## Contribution

The paper develops a new class of higher-order Stein kernels, establishing their properties and applications to enhance convergence rate bounds in the CLT.

## Key findings

- New explicit rates of convergence in the multidimensional CLT.
- Relations between higher-order Stein discrepancies and probability metrics.
- Functional inequalities involving higher-order Stein kernels.

## Abstract

We introduce higher-order Stein kernels relative to the standard Gaussian measure, which generalize the usual Stein kernels by involving higher-order derivatives of test functions. We relate the associated discrepancies to various metrics on the space of probability measures and prove new functional inequalities involving them. As an application, we obtain new explicit improved rates of convergence in the classical multidimensional CLT under higher moment and regularity assumptions.