  We consider the spectral properties of a class of regularized estimators of
(large) empirical covariance matrices corresponding to stationary (but not
necessarily Gaussian) sequences, obtained by banding. We prove a law of large
numbers (similar to that proved in the Gaussian case by Bickel and Levina),
which implies that the spectrum of a banded empirical covariance matrix is an
efficient estimator. Our main result is a central limit theorem in the same
regime, which to our knowledge is new, even in the Gaussian setup.
