Weighted Poincar\'e inequalities, concentration inequalities and tail bounds related to the Stein kernels in dimension one

TL;DR

This paper explores the connections between Stein's density approach and various functional and concentration inequalities in one dimension, introducing new bounds and inequalities related to Stein kernels.

Contribution

It establishes the existence and uniqueness of Stein kernels under certain conditions and derives new weighted Poincaré, log-Sobolev, and concentration inequalities.

Findings

Existence and uniqueness of Stein kernels for measures with finite first moment and connected support.

New weighted Poincaré and log-Sobolev inequalities involving Stein kernels.

Generalized concentration inequalities, including Mills' type and sub-gamma bounds.

Abstract

We investigate links between the so-called Stein's density approach in dimension one and some functional and concentration inequalities. We show that measures having a finite first moment and a density with connected support satisfy a weighted Poincar\'e inequality with the weight being the Stein kernel, that indeed exists and is unique in this case. Furthermore, we prove weighted log-Sobolev and asymmetric Brascamp-Lieb type inequalities related to Stein kernels. We also show that existence of a uniformly bounded Stein kernel is sufficient to ensure a positive Cheeger isoperimetric constant. Then we derive new concentration inequalities. In particular, we prove generalized Mills' type inequalities when a Stein kernel is uniformly bounded and sub-gamma concentration for Lipschitz functions of a variable with a sub-linear Stein kernel. When some exponential moments are finite, a general concentration inequality is then expressed in terms of Legendre-Fenchel transform of the Laplace transform of the Stein kernel. Along the way, we prove a general lemma for bounding the Laplace transform of a random variable, that should be useful in many other contexts when deriving concentration inequalities. Finally, we provide density and tail formulas as well as tail bounds, generalizing previous results that where obtained in the context of Malliavin calculus.