Donsker's theorem in {Wasserstein}-1 distance

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.

References in corpus (4)

• A multivariate Berry--Esseen theorem with explicit constants
• A multivariate central limit theorem for Lipschitz and smooth test functions
• Diffusion approximations via Stein's method and time changes
• Regularity of solutions of the Stein equation and rates in the multivariate central limit theorem

Cited by in corpus (5)

• Stein's method, Gaussian processes and Palm measures, with applications to queueing
• Diffusive limits of Lipschitz functionals of Poisson measures
• Malliavin-Stein Method: a Survey of Recent Developments
• Rate of Convergence in the Functional Central Limit Theorem for Stable Processes
• Quantitative control of Wasserstein distance between Brownian motion and the Goldstein--Kac telegraph process