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

Abstract

In this paper we show that the empirical eigenvalue distribution of any sample covariance matrix generated by independent samples 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, so, the underlying process can exhibit long memory. If the stationary sequence has trivial left sigma field the result holds without any other additional assumptions. This is always true if the entries are functions of i.i.d.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.