Prediction Error for Continuous-Time Stationary Processes with Singular Spectral Densities

The paper considers the mean square linear prediction problem for some classes of continuous-time stationary Gaussian processes with spectral densities possessing singularities. Specifically, we are interested in estimating the rate of decrease to zero of the relative prediction error of a future value of the process using the finite past, compared with the whole past, provided that the underlying process is nondeterministic and is "close" to white noise. We obtain explicit expressions and asymptotic formulae for relative prediction error in the cases where the spectral density possess either zeros (the underlying model is an anti-persistent process), or poles (the model is a long memory processes). Our approach to the problem is based on the Krein's theory of continual analogs of orthogonal polynomials and the continual analogs of Szego theorem on Toeplitz determinants. A key fact is that the relative prediction error can be represented explicitly by means of the so-called "parameter function" which is a continual analog of the Verblunsky coefficients (or reflection parameters) associated with orthogonal polynomials on the unit circle. To this end first we discuss some properties of Krein's functions, state continual analogs of Szego "weak" theorem, and obtain formulae for the resolvents and Fredholm determinants of the corresponding Wiener-Hopf truncated operators.