Skip to main content

The modified, discrete, Levy-transformation is Bernoulli

  • Conference paper
  • First Online:
Séminaire de Probabilités XXVI

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 1526))


From the absolute value of a martingale, X, there is a unique increasing process that can be subtracted so as to obtain a martingale, Y. Paul Levy discovered that if X is Brownian motion, B, then Y, too, is a Brownian motion. Equivalently, Levy found that the transformation that maps B to Y is measure-preserving. Whether it is ergodic, a question raised by Marc Yor, is open. Here, the natural analogue of Levy's transformation for the symmetric random walk is modified and, thus modified, is shown to be measure-preserving. The ergodicity of this transformation is then established by showing that it is isomorphic to the one-sided, Bernoulli shift-transformation associated with a sequence of independent random variables, each uniformly distributed on the unit interval.

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

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  • Doob, J. L., 1953. Stochastic Processes. John Wiley & Sons, Inc., New York.

    MATH  Google Scholar 

  • Levy, Paul, 1939. Sur certains processus stochastiques homogenes. Compositio Mathematica 7, 283–339.

    MathSciNet  MATH  Google Scholar 

  • Levy, Paul, 1948. Processus Stochastiques et Mouvement Brownien. Gauthier-Villars.

    Google Scholar 

  • Kuratowski, K. 1966. Topology. Academic Press, New York and London.

    MATH  Google Scholar 

  • Ornstein, Donald, 1974. Ergodic Theory, Randomness, and Dynamical Systems. Yale University Press.

    Google Scholar 

  • Sinai, Ya.G. 1962. A Weak Isomorphism of Transformations Having an Invariant Measure. Dokl.Akad.Nauk. SSSR 147, 797–800.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations


Editor information

Jacques Azéma Marc Yor Paul André Meyer

Rights and permissions

Reprints and permissions

Copyright information

© 1992 Springer-Verlag

About this paper

Cite this paper

Dubins, L.E., Smorodinsky, M. (1992). The modified, discrete, Levy-transformation is Bernoulli. In: Azéma, J., Yor, M., Meyer, P.A. (eds) Séminaire de Probabilités XXVI. Lecture Notes in Mathematics, vol 1526. Springer, Berlin, Heidelberg.

Download citation

  • DOI:

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-56021-0

  • Online ISBN: 978-3-540-47342-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics