  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),$ $-n\le
t\le-1$, as $n$ goes to infinity. This behavior depends on the regularity
(deterministic or non-deterministic) of the process $X(t)$. In his seminal
paper {\it "Some purely deterministic processes" (J. of Math. and Mech.,} 6(6),
801-810, 1957), M. Rosenblatt has described the asymptotic behavior of the
prediction error for discrete-time deterministic processes in the following two
cases: (a) the spectral density $f(\lambda)$ of $X(t)$ is continuous and
vanishes on an interval, (b) the spectral density $f(\lambda)$ has a very high
order contact with zero. He showed that in the case (a) the prediction error
variance behaves exponentially, while in the case (b), it behaves
hyperbolically as $n\to\infty$. In this paper, using a new approach, we
describe extensions of Rosenblatt's results to broader classes of spectral
densities. Examples illustrate the obtained results.
