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.

No abstract

No abstract

No abstract

In this paper, we obtain necessary as well as sufficient conditions for exponential rate of decrease of the variance of the best linear unbiased estimator (BLUE) for the unknown mean of a stationary sequence possessing a spectral density. In particular, we show that a necessary condition for variance of BLUE to decrease to zero exponentially is that the spectral density vanishes on a set of positive Lebesgue measure in any vicinity of zero.