Rates of convergence in the free central limit theorem

TL;DR

This paper investigates the rate at which sums of free random variables converge to the semicircle distribution in the free central limit theorem, providing new estimates and extending previous results without moment restrictions.

Contribution

It introduces a Kolmogorov distance estimate for normalized sums of free variables and enhances convergence rate results under third moment conditions.

Findings

Established a Kolmogorov distance estimate for free CLT

Proved a free Lindeberg CLT without moment conditions

Improved convergence rates assuming third moments

Abstract

We study the free central limit theorem for not necessarily identically distributed free random variables where the limiting distribution is the semicircle distribution. Starting from an estimate for the Kolmogorov distance between the measure of suitably normalized sums of free random variables and the semicircle distribution without any moment condition, we show the free Lindeberg central limit theorem and improve the known results on rates of convergence under the conditions of the existence of the third moments.