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\footnotesize \noindent Statistica Sinica {\bf 19} (2009), 1755-1768

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\normalsize
 \centerline{\large\bf BANDING SAMPLE AUTOCOVARIANCE}
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 \centerline{\large\bf MATRICES OF STATIONARY PROCESSES}
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 \author{Wei Biao Wu and Mohsen Pourahmadi}
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 \centerline{\it The University of Chicago and Texas A{\rm \&}M University}
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\begin{quotation}
\noindent {\it 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. \par
 \vspace{9pt}
 \noindent
{\it Key words and phrases:} Covariance matrix, prediction,
regularization, short-range dependence, stationary process.
\par
\end{quotation}

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