Structure of stationary finite observation records of discrete-time stochastic linear systems

We discuss the structural features of finite observation records from discrete-time stationary Gaussian stochastic processes with rational power spectra. The processes may be viewed as arising from discrete-time linear systems excited by white noise or as autoregressive-moving average processes. The latter parametrization is chosen for convenience and the existence of nontrivial sufficient statistics is studied. It is shown that only autoregressive processes have sufficient statistics whose dimension is less than the number of observations. Some connections with stochastic realization and nonlinear filtering are described.