Random walk local time approximated by a Wiener sheet combined with an independent Brownian motion

TL;DR

This paper presents a strong approximation of the local time of a symmetric random walk using a Wiener sheet and an independent Brownian motion, providing new insights into the process's behavior.

Contribution

It introduces a novel approximation method for the local time process of a symmetric random walk using Wiener sheets and Brownian motions.

Findings

Strong approximation of local time by Wiener sheet and Brownian motion

Connections between random walk local time and continuous stochastic processes

Implications for understanding local time behavior in stochastic processes

Abstract

Let $\xi(k,n)$ be the local time of a simple symmetric random walk on the line. We give a strong approximation of the centered local time process $\xi(k,n)-\xi(0,n)$ in terms of a Wiener sheet and an independent Wiener process, time changed by an independent Brownian local time. Some related results and consequences are also established.