# Donsker's theorem in {Wasserstein}-1 distance

## TL;DR

This paper establishes bounds on the Wasserstein-1 distance between a random walk and Brownian motion, providing new estimates and applications to convergence rates of local times.

## Contribution

It introduces a novel estimate of the Lipschitz modulus of Stein's equation solution to analyze convergence in Wasserstein-1 distance.

## Key findings

- Derived explicit bounds for Wasserstein-1 distance between random walk and Brownian motion
- Provided a rate of convergence for the local time at zero of Brownian motion
- Developed a new method based on Lipschitz estimates of Stein's equation

## Abstract

We compute the Wassertein-1 (or Kolmogorov-Rubinstein) distance between a random walk in $R^d$ and the Brownian motion. The proof is based on a new estimate of the Lipschitz modulus of the solution of the Stein's equation. As an application, we can evaluate the rate of convergence towards the local time at 0 of the Brownian motion.