Matrix measures, random moments and Gaussian ensembles

TL;DR

This paper investigates the asymptotic behavior of matrix moments in the space of matrix measures, revealing that standardized moments converge to independent Gaussian ensembles, with new relations between moments and canonical moments being key to the analysis.

Contribution

It introduces novel relations between ordinary and canonical moments of matrix measures and demonstrates their asymptotic independence and Gaussian convergence.

Findings

Canonical moments are independent multivariate Beta variables.

These Beta variables converge to Gaussian ensembles as parameters grow.

Standardized moments converge to independent Gaussian ensembles.

Abstract

We consider the moment space $\mathcal{M}_n$ corresponding to $p \times p$ real or complex matrix measures defined on the interval $[0,1]$. The asymptotic properties of the first $k$ components of a uniformly distributed vector $(S_{1,n}, ..., S_{n,n})^* \sim \mathcal{U} (\mathcal{M}_n)$ are studied if $n \to \infty$. In particular, it is shown that an appropriately centered and standardized version of the vector $(S_{1,n}, ..., S_{k,n})^*$ converges weakly to a vector of $k$ independent $p \times p$ Gaussian ensembles. For the proof of our results we use some new relations between ordinary moments and canonical moments of matrix measures which are of their own interest. In particular, it is shown that the first $k$ canonical moments corresponding to the uniform distribution on the real or complex moment space $\mathcal{M}_n$ are independent multivariate Beta distributed random variables and that each of these random variables converge in distribution (if the parameters converge to infinity) to the Gaussian orthogonal ensemble or to the Gaussian unitary ensemble, respectively.