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.
"Limits of renewal processes and Pitman-Yor distribution." Electron. Commun. Probab. 20 1 - 13, 2015. https://doi.org/10.1214/ECP.v20-4080