Prediction error for continuous-time stationary processes with singular spectral densities
This paper deals with the mean square linear prediction problem for some classes of continuous-time stationary Gaussian processes. The spectral density, \(f(\lambda)\), of the underlying process is allowed to possess singularities. The authors define a relative prediction error, \(\delta(f;T)\), of a future value of the process using the past of length \(T\), compared with the whole past. They are interested in estimating the rate of decrease of \(\delta(f;T)\) to zero as \(T \rightarrow \infty\), under the assumption that the underlying process is nondeterministic and ``close'' to white noise. They obtain explicit expressions and asymptotic formulas for \(\delta(f;T)\) in the case where the spectral density possesses either zeros (the underlying model is an antipersistent process), or poles (the model is a long memory process). The results show that the asymptotic relation \(\delta(f;T) \sim 1/T\) as \(T \rightarrow \infty\) is valid whenever the spectral density \(f(\lambda)\) of the underlying process possesses at least one singularity (zero or pole) of power type. The approach to the problem, applied by the authors, is based on the Krein's theory of continual analogs of orthogonal polynomials on the unit circle and the continual analogs of Szegö's theorem on Toeplitz determinants [see, e.g., \textit{A. Böttcher} and \textit{B. Silbermann}, Analysis of Toeplitz operators. Berlin etc.: Springer (1990; Zbl 0732.47029)].
scientific article; zbMATH DE number 5712234
scientific article