We provide a mild sufficient condition for a probability measure on the real line to satisfy a modified log-Sobolev inequality for convex functions, interpolating between the classical log-Sobolev inequality and a Bobkov-Ledoux type inequality. As a consequence we obtain dimension-free two-level concentration results for convex functions of independent random variables with sufficiently regular tail decay. We also provide a link between modified log-Sobolev inequalities for convex functions and weak transport-entropy inequalities, complementing recent work by Gozlan, Roberto, Samson, and Tetali.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm8319-12-2015,
author = {Rados\l aw Adamczak and Micha\l\ Strzelecki},
title = {Modified log-Sobolev inequalities for convex functions on the real line. Sufficient conditions},
journal = {Studia Mathematica},
volume = {231},
year = {2015},
pages = {59-93},
zbl = {1331.60036},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm8319-12-2015}
}
Radosław Adamczak; Michał Strzelecki. Modified log-Sobolev inequalities for convex functions on the real line. Sufficient conditions. Studia Mathematica, Tome 231 (2015) pp. 59-93. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm8319-12-2015/