Donsker's theorem in {Wasserstein}-1 distance

Laure Coutin (1) , Laurent Decreusefond (2, 3, 4)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 INFRES - Département Informatique et Réseaux
3 LTCI - Laboratoire Traitement et Communication de l'Information
4 DIG - Data, Intelligence and Graphs

We compute the Wassertein-1 (or Kantorovitch-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.

Mots clés: Malliavin calculus; Stein method

Domaines: Probabilités [math.PR]

Laure Coutin, Laurent Decreusefond. Donsker's theorem in {Wasserstein}-1 distance. Electronic Communications in Probability, 2020, 25, pp.1--13. ⟨hal-02098892⟩