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In this paper we show that the empirical eigenvalue distribution of anysample covariance matrix generated by independent copies of a stationaryregular sequence has a limiting distribution depending only on the spectraldensity of the sequence. We characterize this limit in terms of Stieltjestransform via a certain simple equation. No rate of convergence to zero of thecovariances is imposed. If the entries of the stationary sequence are functionsof independent random variables the result holds without any other additionalassumptions.  As a method of proof, we study the empirical eigenvalue distribution for asymmetric matrix with independent rows below the diagonal; the entries satisfya Lindeberg-type condition along with mixingale-type conditions without rates.In this nonstationary setting we point out a property of universality, meaningthat, for large matrix size, the empirical eigenvalue distribution depends onlyon the covariance structure of the sequence and is independent on thedistribution leading to it. These results have interest in themselves, allowingto study symmetric random matrices generated by random processes with bothshort and long memory.

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