A State-Space Theory for Stationary Stochastic Processes,

Abstract : Consider a stationary Gaussian stochastic process (y(t);t an element of R) with a rational spectral density, and let H(y) be the Hilbert space spanned by it. The problem of determining all stationary and purely nondeterministic families of minimal splitting subspaces of H(y) is considered; the splitting subspaces constitute state-spaces for the process y. It is shown that some of these families are Markovian, and they lead to internal stochastic realizations. A complete characterization of all Markovian and non-Markovian families of minimal splitting subspaces is provided. Many of the basic results hold without the assumption of rational spectral density. (Author)