Moment estimates implied by modified log-Sobolev inequalities

Adamczak, Radosław; Bednorz, Witold; Wolff, Paweł

doi:10.1051/ps/2016030

Adamczak, Radosław ¹ ; Bednorz, Witold ¹ ; Wolff, Paweł ²

¹ Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland.
² Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland; and Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warszawa, Poland

ESAIM: Probability and Statistics, Tome 21 (2017), pp. 467-494.

Résumé

We study a class of logarithmic Sobolev inequalities with a general form of the energy functional. The class generalizes various examples of modified logarithmic Sobolev inequalities considered previously in the literature. Refining a method of Aida and Stroock for the classical logarithmic Sobolev inequality, we prove that if a measure on $ℝ^{n}$ satisfies a modified logarithmic Sobolev inequality then it satisfies a family of $L^{p}$ -Sobolev-type inequalities with non-Euclidean norms of gradients (and dimension-independent constants). The latter are shown to yield various concentration-type estimates for deviations of smooth (not necessarily Lipschitz) functions and measures of enlargements of sets corresponding to non-Euclidean norms. We also prove a two-level concentration result for functions of bounded Hessian and measures satisfying the classical logarithmic Sobolev inequality.

Reçu le : 2016-05-08
Accepté le : 2016-12-07

MR Zbl

DOI : 10.1051/ps/2016030

Classification : 60E15, 26D10
Mots-clés : Concentration of measure, modified logarithmic Sobolev inequalities

Affiliations des auteurs :

Adamczak, Radosław ¹ ; Bednorz, Witold ¹ ; Wolff, Paweł ²

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     author = {Adamczak, Rados{\l}aw and Bednorz, Witold and Wolff, Pawe{\l}},
     title = {Moment estimates implied by modified {log-Sobolev} inequalities},
     journal = {ESAIM: Probability and Statistics},
     pages = {467--494},
     publisher = {EDP-Sciences},
     volume = {21},
     year = {2017},
     doi = {10.1051/ps/2016030},
     mrnumber = {3743923},
     zbl = {1393.60024},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/ps/2016030/}
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Adamczak, Radosław; Bednorz, Witold; Wolff, Paweł. Moment estimates implied by modified log-Sobolev inequalities. ESAIM: Probability and Statistics, Tome 21 (2017), pp. 467-494. doi : 10.1051/ps/2016030. https://www.numdam.org/articles/10.1051/ps/2016030/

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