On Asymptotic Behavior of Prediction Error for Stationary Generalized Gaussian Processes

The paper addresses the mean square linear prediction problem for a class of stationary generalized Gaussian processes possessing spectral densities. We are particularly interested in the relative prediction error when predicting a future value of the process using a finite past versus using the entire past, given that the underlying process is nondeterministic and ‘‘close’’ to white noise. We establish a necessary and sufficient condition for the relative prediction error to decrease to zero at an exponential rate. Our approach is grounded in Krein’s theory of continuous analogs of orthogonal polynomials. A key fact is that the relative prediction error can be explicitly represented through the so-called ‘‘parameter function,’’ which serves as a continuous analog of the Verblunsky coefficients (or reflection parameters) associated with orthogonal polynomials on the unit circle.