The semicircle law for matrices with independent diagonals

TL;DR

This paper proves that symmetric random matrices with independent diagonals and correlated entries along diagonals have eigenvalue distributions that converge to the semi-circle law, under certain moment conditions.

Contribution

It establishes the semi-circle law for a new class of matrices with correlated entries along diagonals, extending previous results on independent entries.

Findings

Eigenvalue distribution converges to semi-circle law

Convergence holds almost surely under moment conditions

Applicable to matrices with correlated diagonals

Abstract

We investigate the spectral distribution of random matrix ensembles with correlated entries. We consider symmetric matrices with real valued entries and stochastically independent diagonals. Along the diagonals the entries may be correlated. We show that under sufficiently nice moment conditions the empirical eigenvalue distribution converges almost surely weakly to the semi-circle law.