Asymptotic behavior of the prediction error for stationary random sequences (Q1587165)

Let \(\dots ,X(1),X(0),X(-1),\dots\) be a weakly stationary stochastic process with spectral density \(f(\lambda)\). Let \[ \sigma^2_T=\min_{\{a_k\}}E\left[X_0-\sum^T_{k=1}a_kX(-k)\right]^2 \] be the mean square linear prediction error and let \(\sigma^2=\lim\sigma^2_T\) as \(T\to\infty\). The paper investigates the decrease of \(\delta_T=\sigma_T^2-\sigma^2\) to zero as \(T\to\infty\). It is shown that the cases \(\delta_T=O(T^{-\gamma})\) and \(\delta_T=o(T^{-\gamma})\) are possible for \(0<\gamma<1/2\) and various classes of spectral densities \(f(\lambda)\) under restrictions on the type of its zeros. The paper deals also with the asymptotic behavior of Toeplitz determinants \(D_n(f)\) generated by \(f(\lambda)\).