Extensions of Rosenblatt's results on the asymptotic behavior of the prediction error for deterministic stationary sequences

One of the main problem in prediction theory of discrete‐time second‐order stationary processes X(t) is to describe the asymptotic behavior of the best linear mean squared prediction error in predicting X(0) given X(t), −n ≤ t ≤ −1, as n goes to infinity. This behavior depends on the regularity (deterministic or non‐deterministic) of the process X(t). In his seminal article ‘Some purely deterministic processes’ (J. of Math. and Mech., 6(6), 801–10, 1957), Rosenblatt has described the asymptotic behavior of the prediction error for deterministic processes in the following two cases: (i) the spectral density f of X(t) is continuous and vanishes on an interval, (ii) the spectral density f has a very high order contact with zero. He showed that in the case (i) the prediction error behaves exponentially, while in the case (ii), it behaves like a power as n→∞. In this article, using an approach different from the one applied in Rosenblatt's article, we describe extensions of Rosenblatt's results to broader classes of spectral densities. Examples illustrate the obtained results.