Random matrices with exchangeable entries

TL;DR

This paper studies ensembles of symmetric band matrices with exchangeable, correlated entries, showing eigenvalue distributions converge to a semicircle with random scaling and analyzing operator norms, extending de Finetti's theorem.

Contribution

It generalizes de Finetti's theorem to analyze eigenvalue distributions of exchangeable matrices, revealing convergence to a semicircle law with random scaling.

Findings

Eigenvalue measures converge to a semicircle with random scaling.

Correlation structures do not prevent convergence to the semicircle law.

Some results are new even for classical Wigner band matrices.

Abstract

We consider ensembles of real symmetric band matrices with entries drawn from an infinite sequence of exchangeable random variables, as far as the symmetry of the matrices permits. In general the entries of the upper triangular parts of these matrices are correlated and no smallness or sparseness of these correlations is assumed. It is shown that the eigenvalue distribution measures still converge to a semicircle but with random scaling. We also investigate the asymptotic behavior of the corresponding $\ell_2$-operator norms. The key to our analysis is a generalisation of a classic result by de Finetti that allows to represent the underlying probability spaces as averages of Wigner band ensembles with entries that are not necessarily centred. Some of our results appear to be new even for such Wigner band matrices.