Donsker's theorem in {Wasserstein}-1 distance

TL;DR

The paper proves a quantitative Donsker-type result in the Wasserstein-1 distance between a $d$-dimensional random walk and Brownian motion by embedding the problem in fractional Wiener spaces $W_{\eta,p}$ and applying Stein's method with Malliavin calculus. The main advance is a new bound on the Lipschitz modulus of the second derivative of the Stein semigroup, proved via a hierarchy of time discretizations and finite-dimensional projections, which yields an explicit convergence rate $m^{-1/6+\eta/3}\log m$ after balancing scales. As an application, the authors obtain a rate of convergence for the local time at zero of Brownian motion by comparing the Brownian local time to that of the affine interpolation of the random walk. The results provide a pathwise Donsker theorem with explicit, dimensionally robust rates in the Wasserstein-1 metric and illustrate a robust methodological framework combining Stein, Malliavin, and fractional Sobolev/Wiener-space techniques.

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.

Theorems & Definitions (24)

• Lemma 2.1
• proof
• Definition 2.1: Wiener integral
• Definition 2.2: Ornstein-Uhlenbeck semi-group
• Definition 2.3
• Theorem 2.2
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4