Skip to main content
SpringerLink
Log in

Equivalence–Singularity Dichotomy in Markov Measures

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

Abstract

We establish an equivalence–singularity dichotomy for a large class of one-dimensional Markov measures. Our approach is new in that we deal with one-sided and two-sided chains simultaneously and in that we do not appeal to a 0-1 law. In fact, we deduce a new 0-1 law from the dichotomy.

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

Similar content being viewed by others

Data Availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. Avraham-Re’em, N.: On absolutely continuous invariant measures and krieger-type of markov subshifts. Journal d’Analyse Mathématique 147(1), 201–253 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bradley, R. C. Basic properties of strong mixing conditions. a survey and some open questions. Probability surveys 2 (2005), 107–144

  3. Brown, G., and Moran, W. On orthogonality of riesz products. In Mathematical Proceedings of the Cambridge Philosophical Society (1974), vol. 76, Cambridge University Press, pp. 173–181

  4. Chandgotia, N., Han, G., Marcus, B., Meyerovitch, T., Pavlov, R.: One-dimensional markov random fields, markov chains and topological markov fields. Proceedings of the American Mathematical Society 142(1), 227–242 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cohn, H.: On a paper by \(\text{ D }\)oeblin on non-homogeneous markov chains. Advances in Applied Probability 13(2), 388–401 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  6. Danilenko, A., and Lemańczyk, M. K-property for maharam extensions of non-singular bernoulli and markov shifts. Ergodic Theory and Dynamical Systems (2018), 1–30

  7. Danilenko, A., and Silva, C. Ergodic theory: non-singular transformations. Mathematics of Complexity and Dynamical Systems (2011), 329–356

  8. Durrett, R. Probability: theory and examples, vol. 49. Cambridge university press, 2019

  9. Engelbert, H., Shiryaev, A.: On absolute continuity and singularity of probability measures. Banach Cent. Publ 6, 121–132 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  10. Georgii, H. Gibbs measures and phase transitions, vol. 9. Walter de Gruyter, 2011

  11. Grenander, U.: Stochastic processes and statistical inference. Arkiv för matematik 1(3), 195–277 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kabanov, Y., Liptser, R., and Shiryaev, A. Absolute continuity and singularity of locally absolutely continuous probability distributions. i. Mathematics of the USSR-Sbornik 149, 3 (1978), 364–415

  13. Kabanov, Y., Liptser, R., and Shiryaev, A. Absolute continuity and singularity of locally absolutely continuous probability distributions. ii. Mathematics of the USSR-Sbornik 36, 1 (1980), 31

  14. Kakutani, S. On equivalence of infinite product measures. Annals of Mathematics (1948), 214–224

  15. Kosloff, Z.: Proving ergodicity via divergence of time averages. Studia Mathematica 248, 191–215 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  16. LePage, R., Mandrekar, V.: Equivalence-singularity dichotomies from zero-one laws. Proceedings of the American Mathematical Society 31(1), 251–254 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  17. LePage, R., Mandrekar, V.: On likelihood ratios of measures given by markov chains. Proceedings of the American Mathematical Society 52(1), 377–380 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lodkin, A.A.: Absolute continuity of measures corresponding to markov processes with discrete time. Theory of Probability & Its Applications 16(4), 690–694 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pfeiffer, P. Equivalence of totally finite measures on infinite product spaces. Annals of Mathematics (1952), 520–536

  20. Ritter, G.: On kakutani’s dichotomy theorem for infinite products of not necessarily independent functions. Mathematische Annalen 239(1), 35–53 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  21. Sato, H. Absolute continuity of locally equivalent markov chains. Memoirs of the Faculty of Science, Kyushu University. Series A, Mathematics 45, 2 (1991), 285–308

  22. Shiryaev, A. Probability, Graduate Texts in Mathematics, vol. 95. 1995

  23. Shiryaev, A. N. Absolute continuity and singularity of probability measures in functional spaces. In Proceedings of the International Congress of Mathematicians, Helsinki (1978), pp. 209–225

  24. Zachary, S. Countable state space markov random fields and markov chains on trees. The Annals of Probability (1983), 894–903

Download references

Acknowledgements

I thank my advisor, Zemer Kosloff, for his supportive guidance. I also thank Houcein El Abdalaoui and Yair Shenfeld for sending me suggestions and corrections, and the referee for their time and for their useful comments.

Author information

Authors and Affiliations

  1. Einstein Institute of Mathematics, Edmond J. Safra Campus (Givat-Ram), The Hebrew University of Jerusalem, 9190401, Jerusalem, Israel

    Nachi Avraham-Re’em

Authors
  1. Nachi Avraham-Re’em
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nachi Avraham-Re’em.

Ethics declarations

Conflicts of interest

The author has no conflicts of interest to disclose.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Avraham-Re’em, N. Equivalence–Singularity Dichotomy in Markov Measures. J Theor Probab 36, 1437–1453 (2023). https://doi.org/10.1007/s10959-022-01220-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-022-01220-x

Keywords

Mathematics Subject Classification

Search

Navigation