Covariance matrix estimation for stationary time series

Abstract

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.

Published in at http://dx.doi.org/10.1214/11-AOS967 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (7)

• Regularized estimation of large covariance matrices
• Covariance regularization by thresholding
• Optimal rates of convergence for covariance matrix estimation
• Operator norm consistent estimation of large-dimensional sparse covariance matrices
• Asymptotic spectral theory for nonlinear time series
• Central limit theorem for Fourier transforms of stationary processes
• Asymptotic Inference of Autocovariances of Stationary Processes

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