Operator-valued and multivariate free Berry-Esseen theorems

TL;DR

This paper establishes a Berry-Esseen type theorem for the rate of convergence in multivariate free central limit theorems, using operator-valued Cauchy transforms and matrix-valued resolvent sets.

Contribution

It introduces a new approach to estimate convergence rates in multivariate free CLT using operator-valued transforms and resolvent set analysis.

Findings

Derived bounds for convergence speed in free CLT

Extended analysis to non-self-adjoint operators

Developed matrix-valued resolvent set framework

Abstract

We address the question of a Berry-Esseen type theorem for the speed of convergence in a multivariate free central limit theorem. For this, we estimate the difference between the operator-valued Cauchy transforms of the normalized partial sums in an operator-valued free central limit theorem and the Cauchy transform of the limiting operator-valued semicircular element. Since we have to deal with in general non-self-adjoint operators, we introduce the notion of matrix-valued resolvent sets and study the behavior of Cauchy transforms on them.