A note on the convergence of renewal and regenerative processes to a Brownian bridge

TL;DR

This paper presents a direct method to derive a Brownian bridge as a limit of renewal and regenerative processes, bypassing the need for conditioning, and illustrates this with multiple examples.

Contribution

It introduces a novel direct approach to obtain Brownian bridges from renewal processes without conditioning, expanding the theoretical understanding of process convergence.

Findings

Brownian bridge can be obtained directly from renewal processes

The approach simplifies the derivation of limiting processes

Multiple examples demonstrate the method's applicability

Abstract

The standard functional central limit theorem for a renewal process with finite mean and variance, results in a Brownian motion limit. This note shows how to obtain a Brownian bridge process by a direct procedure that does not involve conditioning. Several examples are also considered.