ON ASYMPTOTIC BEHAVIOR OF THE PREDICTION ERROR FOR A CLASS OF DETERMINISTIC STATIONARY SEQUENCES

Abstract

We study the prediction problem for deterministic stationary processes \(X(t)\) possessing spectral density \(f\). We describe the asymptotic behavior of the best linear mean squared prediction error \(\sigma_n^2(f)\) in predicting \(X(0)\) given \( X(t)\), \(-n\le t\le-1\), as \(n\) goes to infinity. We consider a class of spectral densities of the form \(f=f_dg\), where \(f_d\) is the spectral density of a deterministic process that has a very high order contact with zero due to which the Szegő condition is violated, while \(g\) is a nonnegative function that can have arbitrary power type singularities. We show that for spectral densities \(f\) from this class the prediction error \(\sigma_n^2(f)\) behaves like a power as \(n \to \infty \). Examples illustrate the obtained results.