Concentration of information content for convex measures

Abstract

We establish sharp exponential deviation estimates of the information content as well as a sharp bound on the varentropy for the class of convex measures on Euclidean spaces. This generalizes a similar development for log-concave measures in recent work of Fradelizi, Madiman and Wang (2016). In particular, our results imply that convex measures in high dimension are concentrated in an annulus between two convex sets (as in the log-concave case) despite their possibly having much heavier tails. Various tools and consequences are developed, including a sharp comparison result for Rényi entropies, inequalities of Kahane-Khinchine-type for convex measures that extend those of Koldobsky, Pajor and Yaskin (2008) for log-concave measures, and an extension of Berwald's inequality. * Some of these results were announced at ISIT 2016 [24].

On the Entropy Power Inequality for the Rényi Entropy of Order [0, 1]

“…Though the result is implicit in [23], we give a derivation in the appendix for the convenience of the reader. A generalization of the result to s-concave random variables (see [7,12]) is planned to be included in a revised version of [20].…”

On the Equivalence of Statistical Distances for Isotropic Convex Measures

“…The next lemma provides bounds on the var-entropy of s-concave random variables, and was established in [26].…”

“…Proof of Proposition 2.6. The proof of Lemma 2.7 in [26] provides information on the constant c 1 . Precisely, one may take…”

A Combinatorial Approach to Small Ball Inequalities for Sums and Differences

“…There are many different varieties of reverse Hölder inequalities, such as Khinchine inequalities or inequalities relating different L p norms of functions; the survey [7] contains a discussion of several of these classes. In particular, reverse Hölder inequalities have found considerable use in recent years at the interface of probability theory and convex geometry (see, e.g., [32,9,8,10,26,25]), For our purposes, we focus on situations where there exists a constant C(p, q) depending only on q ≥ p such that (E X q ) 1/q ≤ C(p, q)(E X p ) 1/p holds for random variables X in a normed measurable space. In general such an inequality does not hold, but it does hold for random variables with log-concave distributions.…”