Schur recursions, error formulas, and convergence of rational estimators for stationary stochastic sequences

An exact and approximate realization theory for estimation and model filters of second-order stationary stochastic sequences is presented. The properties of -lossless matrices as a unifying framework are used. Necessary and sufficient conditions for the exact realization of an estimation filter and a model filter as a submatrix of a -lossless system are deduced. An extension of the so-called Schur algorithm yields an approximate -lossless realization based on partial past information about the process. The geometric properties of such partial realizations and their convergence are studied. Finally, connections with the Nevanlinna-Pick problem are made, and how the techniques presented constitute a generalization of many aspects of the Levinson-Szegö theory of partial realizations is shown. As a consequence generalized recursive formulas for reproducing kernels and Christoffel-Darboux formulas are obtained. In this paper the scalar case is considered. The matrix case will be considered in a separate publication.