Abstract: We consider estimation of covariance matrices of stationary processes. Under a short-range dependence condition for a wide class of nonlinear processes, it is shown that the banded covariance matrix estimates converge in operator norm to the true covariance matrix with explicit rates of convergence. We also establish the consistency of the estimate of the inverse covariance matrix. These results are applied to a prediction problem, and error bounds for the finite predictor coefficients are obtained. A sub-sampling approach is proposed to choose the banding parameter, and simulation results reveal its satisfactory performance for linear and certain nonlinear processes as the procedure is solely based on the second-order characteristics of the underlying process. Selection of the band parameter for nonlinear processes remains an open problem.
Key words and phrases: Covariance matrix, prediction, regularization, short-range dependence, stationary process.