Electronic Communications in Probability

Limits of renewal processes and Pitman-Yor distribution

Bojan Basrak

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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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465320978

Digital Object Identifier
doi:10.1214/ECP.v20-4080

Mathematical Reviews number (MathSciNet)
MR3374301

Zentralblatt MATH identifier
1327.60080

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

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

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

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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