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

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 $n^{\frac{1}{4}}(\log n{)}^{\frac{3}{4}}$ that is close to the best possible. The tools we apply are almost exclusively from elementary probability theory.

17 pages

References in corpus (1)

• Strong approximation of fractional Brownian motion by moving averages of simple random walks

Cited by in corpus (1)

• Stochastic integration based on simple, symmetric random walks