A characterization of a class of convex log-Sobolev inequalities on the real line

Abstract

We give a sufficient and necessary condition for a probability measure µ on the real line to satisfy the logarithmic Sobolev inequality for convex functions. The condition is expressed in terms of the unique left-continuous and non-decreasing map transporting the symmetric exponential measure onto µ. The main tool in the proof is the theory of weak transport costs.As a consequence, we obtain dimension-free concentration bounds for the lower and upper tails of convex functions of independent random variables which satisfy the convex log-Sobolev inequality.Due to its tensorization property the log-Sobolev inequality is a powerful tool and can be used to obtain dimension-free concentration bounds (via the Herbst argument). It has been investigated also in more general settings of Riemannian manifolds and in context of applications to the study of Markov chains, see e.g. the monographs [3,4] and the expository article [11].

A note on concentration for polynomials in the Ising model

“…Indeed for the right-hand side to be finite ν must be also supported on [−1, 1] n , in which case by (5.3) T θ (µ|ν) = T γ (µ|ν) and T θ (ν|µ) = T γ (ν|µ). In fact, weak transportation inequalities with such strengthened cost functions can hold only for compactly supported measures (see [38]). The interest in such strengthening lies in the fact that by taking into account the boundedness of random variables, it implies concentration inequalities stronger than the subgaussian bound given by (5.2) (see e.g., [4] for concentration results corresponding to various cost functions θ).…”

On the convex infimum convolution inequality with optimal cost function

“…Thanks to the tensorisation property of the ICI, he recovered the Gaussian concentration inequality as well as the so-called Talagrand two-level concentration inequality for the exponential product measure. Moreover, Maurey proved that bounded random variables satisfy the convex ICI with a quadratic cost function (see also Lemma 3.2 in [14] for an improvement).…”

Transport inequalities for random point measures

“…We refer to [21,34,35] for other results connecting transport inequalities of the form (T c ) and variants of the logarithmic Sobolev inequality.…”