Realizations of stationary stochastic processes: applications of passive system theory. (English) Zbl 1289.60065

The authors investigate realizations of a \(p\)-dimensional regular weak stationary discrete-time stochastic process \(y(t)\) as the output data of a passive linear bi-stable discrete-time dynamical system. The state \(x(t)\) is assumed to tend to zero, as \(t\to -\infty\), and the input data is the \(m\)-dimensional white noise. The results are based on the development by the authors of the Darlington method for passive impedance system with losses; see [J. Math. Sci., New York 156, No. 5, 742–760 (2009); translation from Zap. Nauchn. Semin. POMI 355, 37–71 (2008; Zbl 1196.47011)]. It is shown that the realization for a discrete-time process is possible if the spectral density of the process is a nontangential boundary value of a matrix-valued meromorphic function of rank \(m\) with bounded Nevanlinna characteristic in the open unit disk. A classification of such realizations is given; those which can be obtained by Kalman filters are identified. This is a further development of the Lindquist-Picci realization theory [A. Lindquist and G. Picci, SIAM J. Control Optimization 23, 809–857 (1985; Zbl 0593.93048)].


60G10 Stationary stochastic processes
93C55 Discrete-time control/observation systems
37L55 Infinite-dimensional random dynamical systems; stochastic equations
93E11 Filtering in stochastic control theory