IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v19y2006i2d10.1007_s10959-006-0016-3.html
   My bibliography  Save this article

Modified Logarithmic Sobolev Inequalities in Discrete Settings

Author

Listed:
  • Sergey G. Bobkov

    (University of Minnesota)

  • Prasad Tetali

    (School of Mathematics and College of Computing, Georgia Tech)

Abstract

Motivated by the rate at which the entropy of an ergodic Markov chain relative to its stationary distribution decays to zero, we study modified versions of logarithmic Sobolev inequalities in the discrete setting of finite Markov chains and graphs. These inequalities turn out to be weaker than the standard log-Sobolev inequality, but stronger than the Poincare’ (spectral gap) inequality. We show that, in contrast with the spectral gap, for bounded degree expander graphs, various log-Sobolev constants go to zero with the size of the graph. We also derive a hypercontractivity formulation equivalent to our main modified log-Sobolev inequality. Along the way we survey various recent results that have been obtained in this topic by other researchers.

Suggested Citation

  • Sergey G. Bobkov & Prasad Tetali, 2006. "Modified Logarithmic Sobolev Inequalities in Discrete Settings," Journal of Theoretical Probability, Springer, vol. 19(2), pages 289-336, June.
    • Handle: RePEc:spr:jotpro:v:19:y:2006:i:2:d:10.1007_s10959-006-0016-3
    DOI: 10.1007/s10959-006-0016-3
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-006-0016-3
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10959-006-0016-3?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Goel, Sharad, 2004. "Modified logarithmic Sobolev inequalities for some models of random walk," Stochastic Processes and their Applications, Elsevier, vol. 114(1), pages 51-79, November.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Holger Sambale & Arthur Sinulis, 2022. "Concentration Inequalities on the Multislice and for Sampling Without Replacement," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2712-2737, December.
    2. Friedrich Götze & Holger Sambale & Arthur Sinulis, 2021. "Concentration Inequalities for Bounded Functionals via Log-Sobolev-Type Inequalities," Journal of Theoretical Probability, Springer, vol. 34(3), pages 1623-1652, September.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.

      Corrections

      All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jotpro:v:19:y:2006:i:2:d:10.1007_s10959-006-0016-3. See general information about how to correct material in RePEc.

      If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

      If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

      If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

      For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

      Please note that corrections may take a couple of weeks to filter through the various RePEc services.

      IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.