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

Mario Kieburg, Jiyuan Zhang

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 Pólya 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.

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

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 Pólya 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.

Paper Structure

This paper contains 20 sections, 23 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. Introductory Remarks
  3. Main Results and Structure of the Article
  4. Characteristic Functions for Matrices
  5. Invariant Ensembles and their Induced Measures
  6. Fourier and spherical transforms
  7. Derivative principle
  8. Stable Invariant Ensembles
  9. Identifications of Stable Invariant Ensembles
  10. Proofs of Theorem \ref{['p3.1']} and \ref{['p3.1a']}
  11. Examples
  12. Elliptical Stable Ensembles
  13. Dirac Point Mass Spectral Measures
  14. Orbital Spectral Measure
  15. The Generalised Central Limit Theorem
  16. ...and 5 more sections

Key Result

Theorem 1

Let $S_N$ be the symmetric group and $X\in\mathrm{Herm}(N)$ be $\mathrm U(N)$-invariant random matrix with a Borel probability distribution $F$. Its corresponding marginal measures for the unordered eigenvalues is $f_\mathrm{eig}$ and for the unordered diagonal entries it is $f_\mathrm{diag}$. Then, for any Schwartz function $\phi\in \mathscr S(\mathbb R^N)$. The polynomial $\Delta(x)=\prod_{1\leq

Theorems & Definitions (51)

  • Theorem 1: Derivative Principle
  • Theorem 2: Stable Distributions of Invariant Random Matrices
  • Theorem 3: Domains of Attraction
  • Definition 4: Fourier Transform on $L^1(\mathbb R^N)$
  • Definition 5: Spherical Transform on $\mathscr S(\mathbb R^d)$
  • Definition 6: Fourier and Spherical Transform on Tempered Distributions
  • Proposition 7: Derivative Principle for Densities FarautCDKW14MZB16ZK2020
  • proof : Proof of Theorem \ref{['p2.2']}
  • Definition 8: Stable and Strictly Stable Random Matrix Ensembles
  • Theorem 9: Identification with Stable Distributions in $\mathbb R^N$
  • ...and 41 more