On the prediction error for singular stationary processes and transfinite diameters of related sets

The paper is devoted to the prediction problem for discrete-time singular stationary processes with spectral density $f$ and related topics in the case where $f$ vanishes on a set of positive Lebesgue measure. We first discuss the Fekete theorem and its extension due to Robinson on the transfinite diameters of related sets, and prove an extension of Robinson's theorem. For some special sets the transfinite diameters are calculated explicitly by using Robinson's theorem. The obtained results are applied to describe the asymptotic behavior of the prediction error. Then we discuss the Davisson theorem concerning upper bound for the prediction error, and prove its extension. As an application, we obtain estimates for the minimal eigenvalue of a Toeplitz matrix associated with spectral density $f$.