On the empirical spectral distribution for matrices with long memory and independent rows

TL;DR

This paper demonstrates that the empirical spectral distribution of sample covariance matrices from stationary sequences depends solely on their spectral density, with universality properties in nonstationary cases, applicable to matrices with long memory.

Contribution

It characterizes the limiting eigenvalue distribution for matrices with long memory and independent rows, extending previous results to nonstationary and dependent structures.

Findings

Limiting spectral distribution depends only on spectral density.

Universality property in nonstationary settings.

No rate of covariance decay required for results.

Abstract

In this paper we show that the empirical eigenvalue distribution of any sample covariance matrix generated by independent copies of a stationary regular sequence has a limiting distribution depending only on the spectral density of the sequence. We characterize this limit in terms of Stieltjes transform via a certain simple equation. No rate of convergence to zero of the covariances is imposed. If the entries of the stationary sequence are functions of independent random variables the result holds without any other additional assumptions. As a method of proof, we study the empirical eigenvalue distribution for a symmetric matrix with independent rows below the diagonal; the entries satisfy a Lindeberg-type condition along with mixingale-type conditions without rates. In this nonstationary setting we point out a property of universality, meaning that, for large matrix size, the empirical eigenvalue distribution depends only on the covariance structure of the sequence and is independent on the distribution leading to it. These results have interest in themselves, allowing to study symmetric random matrices generated by random processes with both short and long memory.