A rate of convergence when generating stable invariant Hermitian random matrix ensembles

TL;DR

This paper investigates how to generate stable invariant Hermitian random matrix ensembles using sums of specific random matrices, deriving optimal convergence rates in supremum norm and total variation distance.

Contribution

It introduces a method to generate these ensembles from stable vectors and Haar matrices, providing the first explicit convergence rate analysis.

Findings

Derived the rate of convergence in supremum norm for the construction.

Proved the convergence rate is optimal within the class of stable invariant matrices.

Established the rate of convergence in total variation distance.

Abstract

Recently, we have classified Hermitian random matrix ensembles that are invariant under the conjugate action of the unitary group and stable with respect to matrix addition. Apart from a scaling and a shift, the whole information of such an ensemble is encoded in the stability exponent determining the ``heaviness'' of the tail and the spectral measure that describes the anisotropy of the probability distribution. In the present work, we address the question how these ensembles can be generated by the knowledge of the latter two quantities. We consider a sum of a specific construction of identically and independently distributed random matrices that are based on Haar distributed unitary matrices and a stable random vectors. For this construction, we derive the rate of convergence in the supremums norm and show that this rate is optimal in the class of all stable invariant random matrices for a fixed stability exponent. As a consequence we also give the rate of convergence in the total variation distance.