An elementary approach to Brownian local time based on simple, symmetric random walks

TL;DR

This paper introduces a straightforward method to define Brownian local time as the almost sure limit of local times of symmetric random walks, emphasizing elementary probability tools and near-optimal convergence rates.

Contribution

It provides an elementary, almost sure construction of Brownian local time via simple symmetric random walks with a detailed convergence rate.

Findings

Almost sure convergence of local times of random walks to Brownian local time

Joint continuity of the limit in time and space variables

Convergence rate close to the theoretical optimum

Abstract

In this paper we define Brownian local time as the almost sure limit of the local times of a nested sequence of simple, symmetric random walks. The limit is jointly continuous in $(t,x)$. The rate of convergence is $n^{\frac14} (\log n)^{\frac34}$ that is close to the best possible. The tools we apply are almost exclusively from elementary probability theory.