  This paper considers regularizing a covariance matrix of $p$ variables
estimated from $n$ observations, by hard thresholding. We show that the
thresholded estimate is consistent in the operator norm as long as the true
covariance matrix is sparse in a suitable sense, the variables are Gaussian or
sub-Gaussian, and $(\log p)/n\to0$, and obtain explicit rates. The results are
uniform over families of covariance matrices which satisfy a fairly natural
notion of sparsity. We discuss an intuitive resampling scheme for threshold
selection and prove a general cross-validation result that justifies this
approach. We also compare thresholding to other covariance estimators in
simulations and on an example from climate data.
