The limiting spectral distribution in terms of spectral density

TL;DR

This paper establishes the existence and describes the limiting spectral distribution of large symmetric random matrices with correlated entries from stationary random fields, including those with long memory, via an equation for its Stieltjes transform.

Contribution

It provides a general framework for the limiting spectral distribution of matrices with correlated entries, extending to long memory processes and Gaussian fields with spectral density.

Findings

Limiting eigenvalue distribution exists for broad classes of correlated matrices.

The distribution is characterized by an equation for its Stieltjes transform.

Results apply to matrices from Gaussian fields with spectral density and functions of i.i.d. variables.

Abstract

For a large class of symmetric random matrices with correlated entries, selected from stationary random fields of centered and square integrable variables, we show that the limiting distribution of eigenvalue counting measure always exists and we describe it via an equation satisfied by its Stieltjes transform. No rate of convergence to zero of correlations is imposed, therefore the process is allowed to have long memory. In particular, if the symmetrized matrices are constructed from stationary Gaussian random fields which have spectral density, the result of this paper gives a complete solution to the limiting eigenvalue distribution. More generally, for matrices whose entries are functions of independent identically distributed random variables the result also holds.