Simple Stein-method proof of the free central limit theorem

Develop a simple, Stein’s method-based proof of the free central limit theorem that parallels the classical Stein approach (à la Charles Stein, 1972), establishing convergence of standardized sums of freely independent self-adjoint random variables to the semicircular distribution.

Background

The paper surveys prior progress on adapting Stein’s method to non-commutative probability, including approaches via non-commutative Malliavin calculus, free Ornstein–Uhlenbeck semigroups, and quantitative fourth-moment theorems. Despite these advances, a direct Stein-style route to the free central limit theorem (free CLT), mirroring the classical presentation of Stein (1972), has not yet been achieved.

The authors introduce a non-commutative Stein framework and apply it to obtain Berry–Esseen-type bounds and Wasserstein estimates for sums of weakly dependent non-commutative variables, but they emphasize that producing a simple, classical-Stein-style proof of the free CLT itself remains unresolved.