  This paper studies the sparsistency and rates of convergence for estimating
sparse covariance and precision matrices based on penalized likelihood with
nonconvex penalty functions. Here, sparsistency refers to the property that all
parameters that are zero are actually estimated as zero with probability
tending to one. Depending on the case of applications, sparsity priori may
occur on the covariance matrix, its inverse or its Cholesky decomposition. We
study these three sparsity exploration problems under a unified framework with
a general penalty function. We show that the rates of convergence for these
problems under the Frobenius norm are of order $(s_n\log p_n/n)^{1/2}$, where
$s_n$ is the number of nonzero elements, $p_n$ is the size of the covariance
matrix and $n$ is the sample size. This explicitly spells out the contribution
of high-dimensionality is merely of a logarithmic factor. The conditions on the
rate with which the tuning parameter $\lambda_n$ goes to 0 have been made
explicit and compared under different penalties. As a result, for the
$L_1$-penalty, to guarantee the sparsistency and optimal rate of convergence,
the number of nonzero elements should be small: $s_n'=O(p_n)$ at most, among
$O(p_n^2)$ parameters, for estimating sparse covariance or correlation matrix,
sparse precision or inverse correlation matrix or sparse Cholesky factor, where
$s_n'$ is the number of the nonzero elements on the off-diagonal entries. On
the other hand, using the SCAD or hard-thresholding penalty functions, there is
no such a restriction.
