Limits of renewal processes and Pitman-Yor distribution

TL;DR

This paper investigates the asymptotic behavior of renewal processes with dependent steps, showing that scaled steps and age converge to the Pitman-Yor distribution, extending classical theorems in renewal theory.

Contribution

It establishes an invariance principle for dependent renewal steps and their age, linking them to the Pitman-Yor distribution, and extends the Dynkin-Lamperti theorem.

Findings

Scaled renewal steps converge to the Pitman-Yor distribution

Age of the renewal process at time t converges jointly with steps

Results generalize classical renewal process theorems

Abstract

We consider a renewal process with regularly varying stationary and weakly dependent steps, and prove that the steps made before a given time $t$, satisfy an interesting invariance principle. Namely, together with the age of the renewal process at time $t$, they converge after scaling to the Pitman--Yor distribution. We further discuss how our results extend the classical Dynkin--Lamperti theorem.