Berry-Esseen bounds for the multivariate $\mathcal{B}$-free CLT and operator-valued matrices

TL;DR

This paper establishes quantitative Berry-Esseen bounds for the operator-valued free CLT and the distribution of operator-valued matrices, advancing the understanding of convergence rates in operator-valued free probability.

Contribution

It provides the first explicit bounds on the Lévý distance for the operator-valued free CLT and extends these estimates to multivariate and matrix-valued settings.

Findings

First quantitative bounds on the Lévý distance for operator-valued free CLT

Bounds applicable to multivariate and matrix-valued operator distributions

Explicit convergence order estimates in the scalar multivariate case

Abstract

We provide bounds of Berry-Esseen type for fundamental limit theorems in operator-valued free probability theory such as the operator-valued free Central Limit Theorem and the asymptotic behaviour of distributions of operator-valued matrices. Our estimates are on the level of operator-valued Cauchy transforms and the L\'evy distance. We address the single-variable as well as the multivariate setting for which we consider linear matrix pencils and noncommutative polynomials as test functions. The estimates are in terms of operator-valued moments and yield the first quantitative bounds on the L\'evy distance for the operator-valued free Central Limit Theorem. Our results also yield quantitative estimates on joint noncommutative distributions of operator-valued matrices having a general covariance profile. In the scalar-valued multivariate case, these estimates could be passed to explicit bounds on the order of convergence under the Kolmogorov distance.