A note on existence of free Stein kernels

TL;DR

This paper establishes the universal existence of free Stein kernels in free probability, providing explicit constructions and deriving new bounds on convergence rates and characterizations of the semicircular law.

Contribution

It introduces two methods for constructing free Stein kernels and proves their existence universally, unlike in classical probability.

Findings

Free Stein kernels always exist in free probability.

New bounds on convergence rates in the free CLT.

Strengthened characterization of the semicircular law.

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.