  We consider estimation of covariance matrices and their inverses (a.k.a.
precision matrices) for high-dimensional stationary and locally stationary time
series. In the latter case the covariance matrices evolve smoothly in time,
thus forming a covariance matrix function. Using the functional dependence
measure of Wu [Proc. Natl. Acad. Sci. USA 102 (2005) 14150-14154 (electronic)],
we obtain the rate of convergence for the thresholded estimate and illustrate
how the dependence affects the rate of convergence. Asymptotic properties are
also obtained for the precision matrix estimate which is based on the graphical
Lasso principle. Our theory substantially generalizes earlier ones by allowing
dependence, by allowing nonstationarity and by relaxing the associated moment
conditions.
