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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
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    References listed on IDEAS

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    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.
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    Cited by:

    1. 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.
    2. 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.

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