Title: Existence of Stein Kernels under a Spectral Gap, and Discrepancy Bound,
Abstract: We establish existence of Stein kernels for probability measures on
$\mathbb{R}^d$ satisfying a Poincaré inequality, and obtain bounds on the
Stein discrepancy of such measures. Applications to quantitative central limit
theorems are discussed, including a new CLT in Wasserstein distance $W_2$ with
optimal rate and dependence on the dimension. As a byproduct, we obtain a
stability version of an estimate of the Poincaré constant of probability
measures under a second moment constraint. The results extend more generally to
the setting of converse weighted Poincaré inequalities. The proof is based on
simple arguments of calculus of variations.
Further, we establish two general properties enjoyed by the Stein
discrepancy, holding whenever a Stein kernel exists: Stein discrepancy is
strictly decreasing along the CLT, and it controls the skewness of a random
vector.