Large covariance/correlation matrix estimation for long memory temporal data

We consider the estimation of high-dimensional covariance and correlation matrices under slow-decaying temporal dependence. For generalized thresholding estimators, convergence rates are obtained and properties of sparsistency and sign-consistency are established. The impact of temporal dependence on convergence rates is also investigated. An intuitive cross-validation method is proposed for the thresholding parameter selection, which shows good performance in simulations. Convergence rates are also obtained for banding method if the covaraince or correlation matrix is bandable. The considered temporal dependence has longer memory than those in the current literature and has particular implications in analyzing resting-state fMRI data for brain connectivity studies.