Stein's method for rough paths

TL;DR

This paper applies Stein's method to quantify the convergence rate of enriched random walks to enriched Brownian motion within fractional Sobolev spaces, extending classical invariance principles.

Contribution

It introduces a Stein-Dirichlet approach to obtain explicit convergence rates for enriched invariance principles in fractional Sobolev topologies.

Findings

Established convergence rates in fractional Sobolev spaces.

Extended classical invariance principles to enriched processes.

Applied Stein-Dirichlet method for precise probabilistic bounds.

Abstract

The original Donsker theorem says that a standard random walk converges in distribution to a Brownian motion in the space of continuous functions. It has recently been extended to enriched random walks and enriched Brownian motion. We use the Stein-Dirichlet method to precise the rate of this convergence in the topology of fractional Sobolev spaces.