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

Abstract

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.