The asymptotic behavior of the prediction error of a stationary sequence with a spectral density of special type

Let $\{\xi_i\}$ be a stationary in the wide sense regular stochastic process with the spectral density function defined by (2). 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 $\delta_n=\sigma_n^2-\sigma_\infty^2=\sigma_n^2-\sigma^2$, then $\delta_n=O(\frac1n)$ and $\varliminf\limits_{n\to\infty}n\delta_n>0$.