Approximation of the Finite Prediction for a Weakly Stationary Process

Abstract

Let $w$ be the spectral density function of a weakly stationary stochastic process with $w = |h|^2, h$ being an outer function in the upper half plane, and let $\rho^\ast(a) = \operatorname{dist}(e^{ita}h/\bar{h}, H^\infty)$, where $H^\infty$ is the space of boundary functions on $R$ for bounded analytic functions in the upper half plane. It is shown that the standard deviation of the difference between the infinite predictor and the finite predictor from the past of length $T$ does not exceed $\rho^\ast(T)/(1 - \rho^\ast(T))$ times the prediction error of the infinite predictor. Some other estimates relating to the difference between the infinite predictor and the finite predictor are also discussed.