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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.doi:10.1214/ecp.v20-4080 fatcat:plfv2xzewjgh3kotrzgqqgwsfq