Stable distributions and domains of attraction for unitarily invariant Hermitian random matrix ensembles

TL;DR

This paper classifies heavy-tailed, unitarily invariant Hermitian random matrix ensembles using stable distribution theory and harmonic analysis, identifying conditions for their domains of attraction and generalizing key principles.

Contribution

It provides a classification of stable Hermitian matrix ensembles and extends the derivative principle to tempered distributions, advancing understanding of eigenvalue and diagonal entry relations.

Findings

Classification of stable matrix ensembles using harmonic analysis

Necessary and sufficient conditions for domains of attraction

Generalization of the derivative principle to tempered distributions

Abstract

We consider random matrix ensembles on the set of Hermitian matrices that are heavy tailed, in particular not all moments exist, and that are invariant under the conjugate action of the unitary group. The latter property entails that the eigenvectors are Haar distributed and, therefore, factorise from the eigenvalue statistics. We prove a classification for stable matrix ensembles of this kind of matrices represented in terms of matrices, their eigenvalues and their diagonal entries with the help of the classification of the multivariate stable distributions and the harmonic analysis on symmetric matrix spaces. Moreover, we identify sufficient and necessary conditions for their domains of attraction. To illustrate our findings we discuss for instance elliptical invariant random matrix ensembles and P\'olya ensembles, the latter playing a particular role in matrix convolutions. As a byproduct we generalise the derivative principle on the Hermitian matrices to general tempered distributions. This principle relates the joint probability density of the eigenvalues and the diagonal entries of the random matrix.