Estimation of Large Covariance and Precision Matrices from Temporally Dependent Observations

We consider the estimation of large covariance and precision matrices from high-dimensional sub-Gaussian observations with slowly decaying temporal dependence that is bounded by certain polynomial decay rate. The temporal dependence is allowed to be long-range so with longer memory than those considered in the current literature. 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 our motivating example, we study the brain functional connectivity using resting-state fMRI time series data with long-range temporal dependence.