Characterizing innovations realizations for random processes

In this paper we are concerned with the theory of second order (linear) innovations for discrete random processes. We show that of existence of a finite dimensional linear filter realizing the mapping from a discrete random process to innovations is equivalent to a certain semiseparable structure of the covariance sequence of the process. We also show that existence of a finite dimensional realization (linear or nonlinear) of the mapping from a process to its innovations implies that the process have this semiseparable covariance sequence property. In particular, for a stationary random process, the spectral density function must be rational.