Covariance Matrix Estimation for Stationary Time Series

We obtain a sharp convergence rate for banded covariance matrix estimates of stationary processes. A precise order of magnitude is derived for spectral radius of sample covariance matrices. We also consider thresholded covariance matrix estimator that can better characterize sparsity if the true covariance matrix is sparse. As our main tool, we implement Toeplitz (1911)'s idea and relate eigenvalues of covariance matrices to the spectral densities or Fourier transforms of the covariances. We develop a large deviation result for quadratic forms of stationary processes using $m$-dependence approximation, under the framework of causal representation and physical dependence measures.