Random matrices with exchangeable entries

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 ℓ 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.

Random Band and Block Matrices with Correlated Entries

“…Further results in the local regime were achieved in [7] for finite range correlations, in [1] for correlated Gaussian entries and in [8] for more general correlations. For structured random matrices with correlations, see [17] (and references therein) for arbitrary correlations within log-size families of matrix entries, but with independence between these families, [19] for random band matrices with exchangeable entries and [12] for random band matrices with approximately uncorrelated entries.…”

Comorbidity Analysis: Overlapping Semicircles With Wigner Law and Random Matrix Theory

“…Random Matix Theories (RMTs) have grown enormously in fields such as wireless communication theories [4], biology in RNA analysis [5], pure mathematics [6], probability [7], among others [8]. The RMTs [8] are a set of matrices in real symmetric bands with inputs extracted from an infinite sequence of interchangeable random variables, as far as the symmetry of the matrices allows it [9]. The entries of the upper triangular matrices of are correlated and these correlations are not assumed to be small or sparse [10].…”