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Markovian representation of discrete-time stationary stochastic vector processes

In a series of papers Lindquist and Picci [13- 19] have developed an abstract theory of stochastic realization for continuous-time stationary (and stationary increment) Gaussian vector processes, consisting of a basic geometric theory in Hilbert space and a characterization of realizations in terms of Hardy functions. In this paper we present a discrete-time version of this theory and illustrate it by a number of concrete examples. Although some of the results turn out to be straight-forward modifications of the continuous-time ones, obtained by translation from the imaginary axis to the unit circle, there are a number of decidedly nontrivial problems which are unique to the discrete-time setting and which yield new results. Among these are such questions as the definition of past and future, different choices creating different models, and, in particular, certain degeneracies which do not occur in the continuous-time case. One type of degeneracy is related to the singularity of the transition function, another to the concept of invariant directions.