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

Donsker's theorem in {Wasserstein}-1 distance