Realization and approximation of stationary stochastic processes

Abstract : To a multivariate stationary stochastic process, the author associates a scattering matrix S, which measures the interaction between the past and future of the process. This matrix valued function can be viewed as the generalized phase function associated with the spectral density. It determines the density up to congruency only for a completely non-deterministic sequence. Using the theory of Adamjan-Arov-Krein on extensions of Hankel operators, this report establishes that the Hankel operator H sub S determines the Laurent operator L sub S as its unique norm preserving lifting. Employing the Nagy-Foias theory on unitary dilations, or its dual, Lax-Phillips scattering operator model, a realization theory for equivalent classes of stationary sequences with the same density is developed. The minimal equivalence class of Markovian representations is induced by the coprime factorization of the scattering matrix. This presents a unified approach to stochastic and deterministic realization theory, with S as the analog of the frequency response function. To obtain reduced order models, the author approximates the given sequence with a jointly stationary one of a lower dimensional state space, minimizing the distance between the two sequences. (Author)