Abstract
The problem of characterizing a k-dimensional statistic contained in the past of a discrete-time stochastic process y, which allows the best linear least-squares prediction of the future of y, is considered. The solution is provided in terms of the Schmidt pairs and singular values of an infinite matrix, and of the linear innovations of y. In the stationary case, the spectral characteristic of the optimal statistic, and of the corresponding prediction estimate, is obtained. In the case of a rational spectrum, the results are shown to assume a form particularly attractive from the algorithmic point of view. The results admit a straightforward extension to multivariate stochastic processes
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