Summary
In this paper a prediction theory is developed under the general idea that the infinite dimensional covariance matrix is a self-adjoint element in a symmetric Banach algebra. The usual Wiener's spectral factorization method for solving stationary Wiener-Hopf equations has been extended to this algebra. Finally, a theorem for factoring a positive definite covariance matrix into upper and lower triangular factors with similar inverses has been proved.
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Rissanen, J., Barbosa, L. A Factorization problem and the problem of predicting non-stationary vector-valued stochastic processes. Z. Wahrscheinlichkeitstheorie verw Gebiete 12, 255–266 (1969). https://doi.org/10.1007/BF00534844
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DOI: https://doi.org/10.1007/BF00534844