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 satisfies a modified logarithmic Sobolev inequality then it satisfies a family of -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.
Accepté le :
DOI : 10.1051/ps/2016030
Mots clés : Concentration of measure, modified logarithmic Sobolev inequalities
@article{PS_2017__21__467_0,
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 = {http://www.numdam.org/articles/10.1051/ps/2016030/}
}
TY - JOUR AU - Adamczak, Radosław AU - Bednorz, Witold AU - Wolff, Paweł TI - Moment estimates implied by modified log-Sobolev inequalities JO - ESAIM: Probability and Statistics PY - 2017 SP - 467 EP - 494 VL - 21 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2016030/ DO - 10.1051/ps/2016030 LA - en ID - PS_2017__21__467_0 ER -
%0 Journal Article %A Adamczak, Radosław %A Bednorz, Witold %A Wolff, Paweł %T Moment estimates implied by modified log-Sobolev inequalities %J ESAIM: Probability and Statistics %D 2017 %P 467-494 %V 21 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2016030/ %R 10.1051/ps/2016030 %G en %F PS_2017__21__467_0
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. http://www.numdam.org/articles/10.1051/ps/2016030/
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