Strong limit of processes constructed from a renewal proces

TL;DR

This paper constructs a family of processes derived from renewal processes that almost surely converge uniformly to Brownian motion, and it quantifies the convergence rate in a specific case.

Contribution

It introduces a new method to construct processes from renewal processes that converge to Brownian motion and provides a rate of convergence analysis.

Findings

Processes converge almost surely to Brownian motion

Uniform convergence on the unit interval

Explicit convergence rate in a specific case

Abstract

We construct a family of processes, from a renewal process, that have realizations that converge almost surely to the Brownian motion, uniformly on the unit time interval. Finally we compute the rate of convergence in a particular case.