Asymptotic behavior of the prediction error for 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 nondeterministic) and on the dependence structure of the underlying observed process $X(t)$. In this paper we consider this problem both for deterministic and nondeterministic processes and survey some recent results. We focus on the less investigated case - deterministic processes. It turns out that for nondeterministic processes the asymptotic behavior of the prediction error is determined by the dependence structure of the observed process $X(t)$ and the differential properties of its spectral density $f$, while for deterministic processes it is determined by the geometric properties of the spectrum of $X(t)$ and singularities of its spectral density $f$.