Abstract
We study the prediction problem for deterministic stationary processes \(X(t)\) possessing spectral density \(f\). We describe the asymptotic behavior of the best linear mean squared prediction error \(\sigma_n^2(f)\) in predicting \(X(0)\) given \( X(t)\), \(-n\le t\le-1\), as \(n\) goes to infinity. We consider a class of spectral densities of the form \(f=f_dg\), where \(f_d\) is the spectral density of a deterministic process that has a very high order contact with zero due to which the Szegő condition is violated, while \(g\) is a nonnegative function that can have arbitrary power type singularities. We show that for spectral densities \(f\) from this class the prediction error \(\sigma_n^2(f)\) behaves like a power as \(n \to \infty \). Examples illustrate the obtained results.
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The authors would like to thank the anonymous referee for careful review of the manuscript and valuable comments and suggestions.
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BABAYAN, N., GINOVYAN, M. ON ASYMPTOTIC BEHAVIOR OF THE PREDICTION ERROR FOR A CLASS OF DETERMINISTIC STATIONARY SEQUENCES. Acta Math. Hungar. 167, 501–528 (2022). https://doi.org/10.1007/s10474-022-01248-9
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DOI: https://doi.org/10.1007/s10474-022-01248-9
Key words and phrases
- prediction problem
- deterministic stationary process
- singular spectral density
- Rosenblatt’s theorem
- weakly varying sequence