Electronic Communications in Probability

Limits of renewal processes and Pitman-Yor distribution

Bojan Basrak

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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.

Article information

Electron. Commun. Probab., Volume 20 (2015), paper no. 51, 13 pp.

Accepted: 18 July 2015
First available in Project Euclid: 7 June 2016

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Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60G55: Point processes 60G70: Extreme value theory; extremal processes 60F05: Central limit and other weak theorems

Dynkin–Lamperti theorem invariance principle Pitman–Yor distribution point process renewal process regular variation

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Basrak, Bojan. Limits of renewal processes and Pitman-Yor distribution. Electron. Commun. Probab. 20 (2015), paper no. 51, 13 pp. doi:10.1214/ECP.v20-4080. https://projecteuclid.org/euclid.ecp/1465320978

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