Distributions on unbounded moment spaces and random moment sequences

TL;DR

This paper introduces distributions on unbounded moment spaces linked to measures on the real line, connecting them with random matrix ensembles and proving a central limit theorem for associated random vectors.

Contribution

It defines new distributions on unbounded moment spaces and establishes their relation to classical random matrix spectral measures and limiting laws.

Findings

Distributions on unbounded moment spaces are characterized.

Connections to spectral measures of random matrix ensembles are established.

A central limit theorem for random vectors on these spaces is proved.

Abstract

In this paper we define distributions on moment spaces corresponding to measures on the real line with an unbounded support. We identify these distributions as limiting distributions of random moment vectors defined on compact moment spaces and as distributions corresponding to random spectral measures associated with the Jacobi, Laguerre and Hermite ensemble from random matrix theory. For random vectors on the unbounded moment spaces we prove a central limit theorem where the centering vectors correspond to the moments of the Marchenko-Pastur distribution and Wigner's semi-circle law.