  We consider the estimation of large covariance and precision matrices from
high-dimensional sub-Gaussian or heavier-tailed observations with slowly
decaying temporal dependence. The temporal dependence is allowed to be
long-range so with longer memory than those considered in the current
literature. We show that several commonly used methods for independent
observations can be applied to the temporally dependent data. In particular,
the rates of convergence are obtained for the generalized thresholding
estimation of covariance and correlation matrices, and for the constrained
$\ell_1$ minimization and the $\ell_1$ penalized likelihood estimation of
precision matrix. Properties of sparsistency and sign-consistency are also
established. A gap-block cross-validation method is proposed for the tuning
parameter selection, which performs well in simulations. As a motivating
example, we study the brain functional connectivity using resting-state fMRI
time series data with long-range temporal dependence.
