Limiting Spectral Distribution for Wigner Matrices with Dependent Entries

Abstract

In this article, we show the existence of limiting spectral distribution of a symmetric random matrix whose entries come from a stationary Gaussian process with covariances satisfying a summability condition. We provide an explicit description of the moments of the limiting measure. We also show that in some special cases the Gaussian assumption can be relaxed. The description of the limiting measure can also be made via its Stieltjes transform which is characterized as the solution of a functional equation. In two special cases, we get a description of the limiting measure — one as a free product convolution of two distributions, and the other one as a dilation of the Wigner semicircular law.