Non-commutative Stein's Method: Applications to Free Probability and Sums of Non-commutative Variables

TL;DR

This paper develops a non-commutative Stein's method for the semicircular distribution, enabling precise approximation of sums of dependent variables in free probability with improved convergence rates.

Contribution

It introduces a simple non-commutative Stein's method framework and derives new Berry-Esseen bounds and convergence rate improvements for sums of non-commutative variables.

Findings

Established a Berry-Esseen theorem under total variation distance.

Achieved enhanced decay rates under non-commutative Wasserstein distance.

Provided robust convergence results for sums of weakly dependent variables.

Abstract

We present a straightforward formulation of Stein's method for the semicircular distribution, specifically designed for the analysis of non-commutative random variables. Our approach employs a non-commutative version of Stein's heuristic, interpolating between the target and approximating distributions via the free Ornstein-Uhlenbeck semigroup. A key application of this work is to provide a new perspective for obtaining precise estimates of accuracy in the semicircular approximation for sums of weakly dependent variables, measured under the total variation metric. We leverage the simplicity of our arguments to achieve robust convergence results, including: (i) A Berry-Esseen theorem under the total variation distance and (ii) Enhancements in rates of decay under the non-commutative Wasserstein distance towards the semicircular distribution, given adequate high-order moment matching conditions.