Weighted Poincaré inequalities, concentration inequalities and tail bounds related to Stein kernels in dimension one

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 Poincaré 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. More generally, when some exponential moments are finite, the Laplace transform of the random variable of interest is shown to bounded from above by 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 may be of independent interest. We also provide density and tail formulas as well as tail bounds, generalizing previous results that where obtained in the context of Malliavin calculus.