A note on existence of free Stein kernels

Abstract

Stein kernels are a way of comparing probability distributions, defined via integration by parts formulas. We provide two constructions of Stein kernels in free probability. One is given by an explicit formula, and the other via free Poincar\'e inequalities. In particular, we show that unlike in the classical setting, free Stein kernels always exist. As corollaries, we derive new bounds on the rate of convergence in the free CLT, and a strengthening of a characterization of the semicircular law due to Biane.