The discrete-time stochastic realization problem: Minimum variance property of the innovations representation

The linear stochastic realization problem for a time-varying process with a smooth separable covariance is briefly described. It is shown that finding all Markovian realizations of the process is equivalent with finding all solutions to a set of constraints on the state-variances. Introducing a partial ordering on this set of nonnegative definite solutions [viz., ??1 ?? ??2 if ??1 - ??2 is nonnegative definite] it is shown that the smallest solution, obtained with the help of a matrix minimality property, is the unique causal and causally invertible Markovian representation. A stochastic interpretation is given based on the fact that the state of the IR is the filtered estimate of the state of any other model.