Skip to main content
Log in

Finitary isomorphisms of renewal point processes and continuous-time regenerative processes

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We show that a large class of stationary continuous-time regenerative processes are finitarily isomorphic to one another. The key is showing that any stationary renewal point process whose jump distribution is absolutely continuous with exponential tails is finitarily isomorphic to a Poisson point process. We further give simple necessary and sufficient conditions for a renewal point process to be finitarily isomorphic to a Poisson point process. This improves results and answers several questions of Soo [33] and of Kosloff and Soo [19].

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. O. Angel and Y. Spinka, Markov chains with exponential return times are finitary, Ergodic Theory and Dynamical Systems 41 (2021), 2918–2926.

    Article  MathSciNet  Google Scholar 

  2. K. Ball, Poisson thinning by monotone factors, Electronic Communications in Probability 10 (2005), 60–69.

    Article  MathSciNet  Google Scholar 

  3. B. K. Beare and A. A. Toda, Geometrically stopped Markovian random growth processes and Pareto tails, https://arxiv.org/abs/1712.01431.

  4. J. Blanchet and P. Glynn, Uniform renewal theory with applications to expansions of random geometric sums, Advances in Applied Probability 39 (2007), 1070–1097.

    Article  MathSciNet  Google Scholar 

  5. J. Feldman and M. Smorodinsky, Bernoulli flows with infinite entropy, Annals of Mathematical Statistics 42 (1971), 381–382.

    Article  MathSciNet  Google Scholar 

  6. W. Feller, An introduction to probability theory and its applications. Vol. II, John Wiley & Sons, New York-London-Sydney, 1971.

    MATH  Google Scholar 

  7. N. Friedman and D. Ornstein, On isomorphism of weak Bernoulli transformations, Advances in Mathematics 5 (1970), 365–394.

    Article  MathSciNet  Google Scholar 

  8. U. Gabor, On the failure of Ornstein theory in the finitary category, https://arxiv.org/abs/1909.11453

  9. A. E. Holroyd, R. Lyons and T. Soo, Poisson splitting by factors, Annals of Probability 39 (2011), 1938–1982.

    Article  MathSciNet  Google Scholar 

  10. A. E. Holroyd, R. Pemantle, Y. Peres and O. Schramm, Poisson matching, Annales de l’Institut Henri Poincaré Probabilités et Statistiques 45 (2009), 266–287.

    Article  MathSciNet  Google Scholar 

  11. A. E. Holroyd, O. Schramm and D. B. Wilson, Finitary coloring, Annals of Probability 45 (2017), 2867–2898.

    Article  MathSciNet  Google Scholar 

  12. V. V. Kalashnikov, Geometric Sums: Bounds for Rare Events With Applications, Mathematics and its Applications, Vol. 413, Kluwer, Dordrecht, 1997.

    Book  Google Scholar 

  13. S. Kalikow and B. Weiss, Explicit codes for some infinite entropy Bernoulli shifts, Annals of Probability 20 (1992), 397–402.

    Article  MathSciNet  Google Scholar 

  14. O. Kallenberg, Foundations of Modern Probability, Probability and its Applications (New York), Springer, New York, 2002.

    Book  Google Scholar 

  15. M. Keane and M. Smorodinsky, A class of finitary codes, Israel Journal of Mathematics 26 (1977), 352–371.

    Article  MathSciNet  Google Scholar 

  16. M. Keane and M. Smorodinsky, Bernoulli schemes of the same entropy are finitarily isomorphic, Annals of Mathematics 109 (1979), 397–406.

    Article  MathSciNet  Google Scholar 

  17. M. Keane and M. Smorodinsky, Finitary isomorphisms of irreducible Markov shifts, Israel Journal of Mathematics 34 (1979), 281–286.

    Article  MathSciNet  Google Scholar 

  18. A. N. Kolmogorov, Entropy per unit time as a metric invariant of automorphisms, Doklady Akademii Nauk SSSR 124 (1959), 754–755.

    MathSciNet  MATH  Google Scholar 

  19. Z. Kosloff and T. Soo, Finitary isomorphisms of Brownian motions, Annals of Probability b (2020), 1966–1979.

  20. D. Ornstein, Bernoulli shifts with the same entropy are isomorphic, Advances in Mathematics 4 (1970), 337–352.

    Article  MathSciNet  Google Scholar 

  21. D. Ornstein, Factors of Bernoulli shifts are Bernoulli shifts, Advances in Mathematics 5 (1970), 349–364.

    Article  MathSciNet  Google Scholar 

  22. D. Ornstein, Imbedding Bernoulli shifts in flows, in Contributions to Ergodic Theory and Probability, Springer, Berlin, 1970, pp. 178–218.

    Chapter  Google Scholar 

  23. D. Ornstein, Two Bernoulli shifts with infinite entropy are isomorphic, Advances in Mathematics 5 (1970), 339–348.

    Article  MathSciNet  Google Scholar 

  24. D. Ornstein, The isomorphism theorem for Bernoulli flows, Advances in Mathematics 10 (1973), 124–142.

    Article  MathSciNet  Google Scholar 

  25. D. Ornstein, Newton’s laws and coin tossing, Notices of the American Mathematical Society 60 (2013). 450–459.

    Article  MathSciNet  Google Scholar 

  26. D. Ornstein and P. Shields, Mixing Markov shifts of kernel type are Bernoulli, Advances in Mathematics 10 (1973), 143–146.

    Article  MathSciNet  Google Scholar 

  27. S. I. Resnick, Adventures in Stochastic Processes, Birkhäuser, Boston, MA, 1992.

    MATH  Google Scholar 

  28. D. J. Rudolph, A characterization of those processes finitarily isomorphic to a Bernoulli shift, in Ergodic Theory and Dynamical Systems. I, Progress in Mathematics, Vol. 10, Birkhauser, Boston, MA, 1981, pp. 1–64.

    Chapter  Google Scholar 

  29. D. J. Rudolph, A mixing Markov chain with exponentially decaying return times is finitarily Bernoulli, Ergodic Theory and Dynamical Systems 2 (1982), 85–97.

    Article  MathSciNet  Google Scholar 

  30. J. Serafin, Finitary codes, a short survey, in Dynamics & Stochastics, Institute of Mathematical Statistics Lecture Notes—Monograph Series, Vol. 48, Institute of Mathematical Statistics, Beachwood, OH, 2006, pp. 262–273.

    Chapter  Google Scholar 

  31. S. M. Shea, Finitary isomorphism of some renewal processes to Bernoulli schemes, Indagationes Mathematicae 20 (2009), 463–476.

    Article  MathSciNet  Google Scholar 

  32. J. Sinai, On the concept of entropy for a dynamic system, Doklady Akademii Nauk SSSR 124 (1959), 768–771.

    MathSciNet  MATH  Google Scholar 

  33. T. Soo, Finitary isomorphisms of some infinite entropy Bernoulli flows, Israel Journal of Mathematics 232 (2019), 883–897.

    Article  MathSciNet  Google Scholar 

  34. T. Soo and A. Wilkens, Finitary isomorphisms of Poisson point processes, Annals of Probability 47 (2019), 3055–3081.

    Article  MathSciNet  Google Scholar 

  35. S. M. Srivastava, A Course on Borel Sets, Graduate Texts in Mathematics, Vol. 180, Springer, New York, 1998.

    Book  Google Scholar 

Download references

Acknowledgments

I am grateful to Nishant Chandgotia, Zemer Kosloff and Edwin Perkins for helpful discussions and comments.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yinon Spinka.

Additional information

This work was supported in part by NSERC of Canada.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Spinka, Y. Finitary isomorphisms of renewal point processes and continuous-time regenerative processes. Isr. J. Math. 249, 857–897 (2022). https://doi.org/10.1007/s11856-022-2328-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-022-2328-0

Navigation