White-Noise Representations in Stochastic Realization Theory

Abstract

It is proved that any random sequence can be exhibited as the output of a stochastic dynamical system driven by white noise. Further refinements are obtained for Markov, stationary, and ergodic sequences. This settles some open problems in stochastic realization theory posed by Willems and Van Schuppen [NATO ASI-AMS Seminar on Algebraic and Geometric Methods in Linear System Theory, Harvard University, Cambridge, MA, 1979].