Stein's method for Brownian approximations

TL;DR

This paper applies Stein's method to analyze convergence rates in Brownian motion approximations, providing explicit bounds and Edgeworth expansions for various limit theorems in probability.

Contribution

It develops a framework using Stein's method to bound Wasserstein distances in infinite-dimensional spaces and derives explicit convergence rates for key probabilistic approximations.

Findings

Poisson approximation of Brownian motion converges at rate proportional to λ^{-1/2}

Speed of convergence for Donsker's theorem is quantified

Edgeworth expansions with precise error bounds are established

Abstract

Motivated by a theorem of Barbour, we revisit some of the classical limit theorems in probability from the viewpoint of the Stein method. We setup the framework to bound Wasserstein distances between some distributions on infinite dimensional spaces. We show that the convergence rate for the Poisson approximation of the Brownian motion is as expected proportional to $\lambda^{-1/2}$ where $\lambda$ is the intensity of the Poisson process. We also exhibit the speed of convergence for the Donsker Theorem and for the linear interpolation of the Brownian motion. By iterating the procedure, we give Edgeworth expansions with precise error bounds.