## Electronic Communications in Probability

### Limits of renewal processes and Pitman-Yor distribution

Bojan Basrak

#### 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

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

Rights

#### 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

#### References

• Basrak, Bojan; KrizmaniÄ‡, Danijel; Segers, Johan. A functional limit theorem for dependent sequences with infinite variance stable limits. Ann. Probab. 40 (2012), no. 5, 2008–2033.
• Basrak, Bojan; Segers, Johan. Regularly varying multivariate time series. Stochastic Process. Appl. 119 (2009), no. 4, 1055–1080.
• Basrak B, and Å poljariÄ‡ D., On randomly spaced observations and continuous time random walks, ARXIVmath.PR/1507.00191.
• Bertoin, Jean. Subordinators: examples and applications. Lectures on probability theory and statistics (Saint-Flour, 1997), 1–91, Lecture Notes in Math., 1717, Springer, Berlin, 1999.
• Billingsley, Patrick. Convergence of probability measures. Second edition. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999. x+277 pp. ISBN: 0-471-19745-9
• Bingham, N. H.; Goldie, C. M.; Teugels, J. L. Regular variation. Encyclopedia of Mathematics and its Applications, 27. Cambridge University Press, Cambridge, 1989. xx+494 pp. ISBN: 0-521-37943-1
• CsÃ¡ki, Endre; Hu, Yueyun. Lengths and heights of random walk excursions. Discrete random walks (Paris, 2003), 45–52 (electronic), Discrete Math. Theor. Comput. Sci. Proc., AC, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2003.
• Jacod, Jean; Shiryaev, Albert N. Limit theorems for stochastic processes. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 288. Springer-Verlag, Berlin, 2003. xx+661 pp. ISBN: 3-540-43932-3
• KrizmaniÄ‡ D., Functional limit theorems for weakly dependent regularly varying time series., Ph.D. thesis, University of Zagreb, 2012, Available at rlhttp://www.math.uniri.hr/~dkrizmanic/DKthesis.pdf.
• Kyprianou, Andreas E. Fluctuations of Lévy processes with applications. Introductory lectures. Second edition. Universitext. Springer, Heidelberg, 2014. xviii+455 pp. ISBN: 978-3-642-37631-3; 978-3-642-37632-0
• Leadbetter, M. R.; Lindgren, Georg; Rootzén, Holger. Extremes and related properties of random sequences and processes. Springer Series in Statistics. Springer-Verlag, New York-Berlin, 1983. xii+336 pp. ISBN: 0-387-90731-9
• O'Brien, George L. Extreme values for stationary and Markov sequences. Ann. Probab. 15 (1987), no. 1, 281–291.
• Perman, Mihael. Random discrete distributions derived from subordinators. Thesis (Ph.D.)â€“University of California, Berkeley. ProQuest LLC, Ann Arbor, MI, 1990. 80 pp.
• Perman, Mihael. Order statistics for jumps of normalised subordinators. Stochastic Process. Appl. 46 (1993), no. 2, 267–281.
• Perman, Mihael; Pitman, Jim; Yor, Marc. Size-biased sampling of Poisson point processes and excursions. Probab. Theory Related Fields 92 (1992), no. 1, 21–39.
• Pitman, Jim; Yor, Marc. Arcsine laws and interval partitions derived from a stable subordinator. Proc. London Math. Soc. (3) 65 (1992), no. 2, 326–356.
• Pitman, Jim; Yor, Marc. The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25 (1997), no. 2, 855–900.
• Resnick, Sidney I. Heavy-tail phenomena. Probabilistic and statistical modeling. Springer Series in Operations Research and Financial Engineering. Springer, New York, 2007. xx+404 pp. ISBN: 978-0-387-24272-9; 0-387-24272-4
• Resnick, Sidney I. Extreme values, regular variation and point processes. Reprint of the 1987 original. Springer Series in Operations Research and Financial Engineering. Springer, New York, 2008. xiv+320 pp. ISBN: 978-0-387-75952-4
• Seneta, Eugene. Regularly varying functions. Lecture Notes in Mathematics, Vol. 508. Springer-Verlag, Berlin-New York, 1976. v+112 pp.
• Teh, Yee Whye; Jordan, Michael I. Hierarchical Bayesian nonparametric models with applications. Bayesian nonparametrics, 158–207, Camb. Ser. Stat. Probab. Math., Cambridge Univ. Press, Cambridge, 2010.
• Whitt, Ward. Stochastic-process limits. An introduction to stochastic-process limits and their application to queues. Springer Series in Operations Research. Springer-Verlag, New York, 2002. xxiv+602 pp. ISBN: 0-387-95358-2