 
Summary: The Annals of Statistics
2008, Vol. 36, No. 6, 25532576
DOI: 10.1214/07AOS503
© Institute of Mathematical Statistics, 2008
A CLT FOR REGULARIZED SAMPLE COVARIANCE MATRICES
BY GREG W. ANDERSON AND OFER ZEITOUNI1
University of Minnesota
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.
1. Introduction. We consider in this paper the spectral properties of a class of
regularized estimators of (large) covariance matrices. More precisely, let X = X(p)
be a data matrix of n independent rows, with each row being a sample of length p
from a mean zero stationary sequence {Zj } whose covariance sequence satisfies
appropriate regularity conditions (for details on those, see Assumption 2.2). Let
XT X denote the empirical covariance matrix associated with the data. We recall
