Estimation of Covariance Matrices and Their Functionals for High-Dimensional Linear Processes

We consider the structured covariance matrix estimation and related problems for high-dimensional linear processes. A comprehensive characterization is given for the spectral and the Frobenius norm rates of convergence for the thresholded covariance matrix estimates. Asymptotic rate of the graphical Lasso estimate of precision matrices is derived for Gaussian processes. In addition, we propose a Dantzig-type estimate of the linear statistic of the form $\Sigma^{-1} \vb$, where $\Sigma$ is a covariance matrix and $\vb$ is a vector. Our framework covers the broad regime from i.i.d. samples to long-range dependent time series and from sub-Gaussian innovations to those with mild polynomial moments. The rate of convergence depends on the degree of temporal dependence and the moment conditions in a subtle way. A data-driven method is proposed for selecting the tuning parameters. Our procedure is applied to a real fMRI dataset.