# Stein's method for normal approximation in Wasserstein distances with application to the multivariate Central Limit Theorem

## TL;DR

This paper develops Stein's method to bound Wasserstein distances for normal approximation, providing optimal convergence rates for the multivariate CLT under minimal moment conditions.

## Contribution

It introduces a novel approach using stochastic processes to bound Wasserstein distances of any order, extending Stein's method for multivariate normal approximation.

## Key findings

- Bounds Wasserstein distance of order 2 using stochastic process
- Extends bounds to Wasserstein distances of any order p ≥ 1
- Provides optimal convergence rates for multivariate CLT

## Abstract

We use Stein's method to bound the Wasserstein distance of order $2$ between a measure $\nu$ and the Gaussian measure using a stochastic process $(X_t)_{t \geq 0}$ such that $X_t$ is drawn from $\nu$ for any $t > 0$. If the stochastic process $(X_t)_{t \geq 0}$ satisfies an additional exchangeability assumption, we show it can also be used to obtain bounds on Wasserstein distances of any order $p \geq 1$. Using our results, we provide optimal convergence rates for the multi-dimensional Central Limit Theorem in terms of Wasserstein distances of any order $p \geq 2$ under simple moment assumptions.