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Moment Estimation Implied by the Bobkov-Ledoux Inequality

  • Witold BednorzEmail author
  • Grzegorz Głowienko
Conference paper
Part of the Progress in Probability book series (PRPR, volume 74)

Abstract

In this paper we consider a probability measure on the high dimensional Euclidean space satisfying Bobkov-Ledoux inequality. Bobkov and Ledoux have shown in (Probab Theory Related Fields 107(3):383–400, 1997) that such entropy inequality captures concentration phenomenon of product exponential measure and implies Poincaré inequality. For this reason any measure satisfying one of those inequalities shares the same concentration result as the exponential measure. In this paper using B-L inequality we derive some bounds for exponential Orlicz norms for any locally Lipschitz function. The result is close to the question posted by Adamczak and Wolff in (Probab Theory Related Fields 162:531–586, 2015) regarding moments estimate for locally Lipschitz functions, which is expected to result from B-L inequality.

Keywords

Concentration of measure Poincaré inequality Sobolev inequality 

Subject Classification

60E15 46N30 

Notes

Acknowledgement

This research was partially supported by NCN Grant UMO-2016/21/B/ ST1/01489.

References

  1. 1.
    R. Adamczak, P. Wolff, Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order. Probab. Theory Related Fields 162, 531–586 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    S. Aida, D. Stroock, Moment estimates derived from Poincare and logarithmic Sobolev inequality. Math. Res. Lett. 1, 75–86 (1994)MathSciNetCrossRefGoogle Scholar
  3. 3.
    S. Bobkov, M. Ledoux, Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution. Probab. Theory Related Fields 107(3), 383–400 (1997)MathSciNetCrossRefGoogle Scholar
  4. 4.
    E. Milman, On the role of convexity in isoperimetry, spectral gap and concentration. Invent. Math. 177, 1–4 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of WarsawWarszawaPoland

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