  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 a thresholded covariance
matrix estimator that can better characterize sparsity if the true covariance
matrix is sparse. As our main tool, we implement Toeplitz [Math. Ann. 70 (1911)
351-376] 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.
