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ON ASYMPTOTIC BEHAVIOR OF THE PREDICTION ERROR FOR A CLASS OF DETERMINISTIC STATIONARY SEQUENCES

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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Acknowledgements

The authors would like to thank the anonymous referee for careful review of the manuscript and valuable comments and suggestions.

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Correspondence to M. GINOVYAN.

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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

Mathematics Subject Classification

  • 60G10
  • 60G25
  • 62M20
  • 62M15