Rate of convergence of the conditioned random walk towards the Brownian bridge

TL;DR

This paper investigates how quickly two discrete processes, a conditioned random walk and an empirical process, converge to the Brownian bridge, using advanced probabilistic methods to quantify the convergence rate.

Contribution

It introduces a novel combination of Stein's method and Radon-Nikodym representation to bound the convergence rate of these processes to the Brownian bridge.

Findings

Bound the Fortet-Mourier distance between processes and the Brownian bridge

Quantitative convergence rates established for conditioned random walk and empirical process

Methodology applicable to other conditioned stochastic processes

Abstract

We study the rate of convergence of two discrete processes towards the Brownian bridge: the random walk conditioned to be zero at time 2n and the empirical process which appears in the Glivencko-Cantelli theorem. Combining a functional Stein method with a Radon-Nikodym representation of the bridge, we bound the Fortet-Mourier distance between these conditioned processes and the Brownian bridge.