Estimating covariances of locally stationary processes: consistency of best basis methods

Mallat, Papanicolaou and Zhang [1995] have suggested a method for approximating the covariance of a locally stationary process by a covariance which is diagonal in an ideally constructed Coifman-Meyer [1991] basis of cosine packets. A natural question arising from their work is to translate approximation results into estimation results. In this paper we discuss the problem of estimation of the covariance from sampled data. We show that it is possible to obtain an empirical basis from sampled data which is nearly as good as the ideal theoretical basis. We describe a specific method which is nicely suited to the Coifman-Wickerhauser [1992] fast algorithm for obtaining a best basis. In this note we describe theoretical results.