Results for convergence rates associated with renewal processes

TL;DR

This paper proves that renewal equations with spread out interarrival times exhibit power law decay, and establishes convergence in distribution for renewal process recurrence times using coupling and regenerative properties.

Contribution

It introduces a coupling-based proof of power law decay in renewal equations and demonstrates convergence in distribution for renewal process recurrence times.

Findings

Power law decay rate for renewal equations with spread out interarrival times

Convergence in distribution of renewal process recurrence times

Results hold under mild moment assumptions

Abstract

Via a coupling argument, it is proved that the solution to a renewal equation has a power law decay rate in the case of a spread out interarrival distribution. By the regenerative property, the convergence in distribution for the recurrence times of the renewal process is established as well as that of the compensator of the renewal counting process. All results are proved under mild assumptions of existence of moments.