On the Asymptotic Behaviour of the Prediction Error

Let $\{x_j\}$ be a stationary stochastic process in the wide sense which is regular, with spectral density function $f(\lambda)$. Denote by $\sigma_n^2$ the mean square prediction error in predicting $x_0$ by linear forms in $x_{-1},x_{-2},\dots,x_{-n}$. Let $\delta_n=\sigma_n^2-\sigma_\infty^2=\sigma_n^2-\sigma^2$. The rate of convergence $\delta_n\downarrow 0$ is investigated in this article.