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Extensions of Rosenblatt’s results on the asymptotic behavior of the prediction error for deterministic stationary sequences. (English) Zbl 1489.60052

Summary: 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 \(X(t), - n \leq t \leq -1\), 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. Math. Mech. 6, 801–810 (1957; Zbl 0080.35001)], M. 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 \(n \to \infty\). 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.

MSC:

60G10 Stationary stochastic processes
60G25 Prediction theory (aspects of stochastic processes)
62M15 Inference from stochastic processes and spectral analysis
62M20 Inference from stochastic processes and prediction

Citations:

Zbl 0080.35001
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References:

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