Stochastic approximation and realization of Hankel matrices

Dynamic systems are considered whose outputs can be represented either by a deterministic series of the input variables or by a stochastic series of brownian motion processes. Approximate representations of such series are proposed. The representations are based on the Hankel matrix representation of the series. First, it is shown that any Hankel matrix, belonging to suitable classes, can be approximated in the deterministic and stochastic sense by finite rank Hankel matrices. Then, a method is proposed to design bilinear realizations of finite rank Hankel matrices, effective in the deterministic and stochastic sense.