Asymptotic behavior of the prediction error for stationary sequences
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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 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.
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