Skip to main content
SpringerLink
Log in

Ages of Records in Random Walks

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We consider random walks with continuous and symmetric step distributions. We prove universal asymptotics for the average proportion of the age of the kth longest lasting record for \(k=1,2,\ldots \) and for the probability that the record of the kth longest age is broken at step n. Due to the relation to the Chinese restaurant process, the ranked sequence of proportions of ages converges to the Poisson–Dirichlet distribution.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Andersen, E.S.: On the fluctuations of sums of random variables I. Math. Scand. 1, 263–285 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barsak, B.: Limits of renewal processes and Pitman-Yor distribution. Electron. Commun. Probab. 20(51), 1–13 (2015)

    MathSciNet  Google Scholar 

  3. Csáki, E., Hu Y.: Lengths and heights of random walk excursions. Discret. Math. Theor. Comput. Sci. AC, 45–52 (2003)

  4. Csáki, E., Hu, Y.: Invariance principles for ranked excursion lengths and heights. Electron. Commun. Probab. 9, 14–21 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Feller, W.: An Introduction to Probability Theory and Its Applications, 2nd edn. Wiley, New York (1971)

    MATH  Google Scholar 

  6. Finch, S.: Excursion durations. unpublished manuscript (2008)

  7. García-García, R., Rosso, A., Schehr, G.: The longest excursion of fractional Brownian motion: numerical evidence of the non-Markovian effects. Phys. Rev. E 81, 010102(R) (2010)

    Article  ADS  Google Scholar 

  8. Godrèche, C.: Longest interval between zeros of the tied-down random walk, the Brownian bridge and related renewal processes. Preprint, arXiv: 1611.01434

  9. Godrèche, C., Majumdar, S.N., Schehr, G.: The longest excursion of stochastic processes in nonequilibrium systems. Phys. Rev. Lett. 102, 240602 (2009)

    Article  ADS  Google Scholar 

  10. Godrèche, C., Majumdar, S.N., Schehr, G.: Universal statistics of longest lasting records of random walks and Lévy flights. J. Phys. A 47, 255001 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Godrèche, C., Majumdar, S.N., Schehr, G.: Statistics of the longest interval in renewal processes. J. Stat. Mech. 2015(3), P03014 (2015)

    Article  MathSciNet  Google Scholar 

  12. Majumdar, S.N., Ziff, R.M.: Universal record statistics of random walks and Lévy flights. Phys. Rev. Lett. 101, 050601 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Perman, M.: Random discrete distributions derived from subordinators. PhD Thesis, University of California, Berkeley (1990)

  14. Perman, M.: Order statistics for jumps of normalised subordinators. Stoch. Process. Appl. 46, 264–281 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  15. Perman, M., Pitman, J., Yor, M.: Size-biased sampling of Poisson point processes and excursions. Probab. Theory Relat. Fields 92, 21–39 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. Pitman, J.: Exchangeable and partially exchangeable random partitions. Probab. Theory Relat. Fields 102, 145–158 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  17. Pitman, J.: Partition structures derived from Brownian motion and stable subordinators. Bernoulli 3, 79–96 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Pitman, J.: Combinatorial Stochastic Processes. Lecture Notes for St. Flour course (2002)

  19. Pitman, J., Yor, M.: Arcsine laws and interval partitions derived from a stable subordinator. Proc. Lond. Math. Soc. 65, 326–356 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  20. Pitman, J., Yor, M.: The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25, 855–900 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  21. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, New York (1999)

    Book  MATH  Google Scholar 

  22. Scheffer, C.L.: The rank of the present excursion. Stoch. Process. Appl. 55, 101–118 (1995)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank Balázs Ráth for discussions related to this paper, in particular for the remark after Theorem 2. They are grateful for the anonymous referees for their insightful suggestions and comments. This work was supported by OTKA (Hungarian National Research Fund) Grant K100473. B. V. is grateful for the Postdoctoral Fellowship of the Hungarian Academy of Sciences and for the Bolyai Research Scholarship.

Author information

Authors and Affiliations

  1. University of Groningen, Nijenborgh 9, 9747 AG, Groningen, The Netherlands

    Réka Szabó

  2. MTA–BME Stochastics Research Group, Egry J. u. 1, Budapest, 1111, Hungary

    Bálint Vető

Authors
  1. Réka Szabó
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bálint Vető
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bálint Vető.

Additional information

Dedicated to Bálint Tóth on the occasion of his 60th birthday.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Szabó, R., Vető, B. Ages of Records in Random Walks. J Stat Phys 165, 1086–1101 (2016). https://doi.org/10.1007/s10955-016-1671-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-016-1671-0

Keywords

Mathematics Subject Classification

Search

Navigation