Canonical moments and random spectral measures

TL;DR

This paper explores the relationship between random moment problems and random matrix theory, focusing on spectral measures derived from classical ensembles and their large deviations properties.

Contribution

It introduces a method to generate spectral measures from random moments and analyzes their large deviations, connecting moment problems with spectral measures in random matrices.

Findings

Spectral measures can be obtained from uniform random moments.

Large deviations of these measures involve the reversed Kullback-Leibler divergence.

The approach links random moment problems with spectral properties of classical random matrices.

Abstract

We study some connections between the random moment problem and the random matrix theory. A uniform draw in a space of moments can be lifted into the spectral probability measure of the pair (A,e) where A is a random matrix from a classical ensemble and e is a fixed unit vector. This random measure is a weighted sampling among the eigenvalues of A. We also study the large deviations properties of this random measure when the dimension of the matrix grows. The rate function for these large deviations involves the reversed Kullback information.