  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.
