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We consider a renewal process with regularly varying stationary and weaklydependent steps, and prove that the steps made before a given time $t$, satisfyan interesting invariance principle. Namely, together with the age of therenewal process at time $t$, they converge after scaling to the Pitman--Yordistribution. We further discuss how our results extend the classicalDynkin--Lamperti theorem.

Limits of renewal processes and Pitman-Yor distribution