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

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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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References

  1. N. M. Babayan, On the asymptotic behavior of prediction error, J. Soviet Math., 27 (1984), 3170–3181.

  2. N. M. Babayan, On asymptotic behavior of the prediction error in the singular case, Theory Probab. Appl., 29 (1984), 147–150.

  3. N. M. Babayan and M. S. Ginovyan, On hyperbolic decay of prediction error variance for deterministic stationary sequences, J. Cont. Math. Anal., 55 (2020), 76–95.

  4. N. M. Babayan, M. S. Ginovyan and M. S. Taqqu, Extensions of Rosenblatt’s results on the asymptotic behavior of the prediction error for deterministic stationary sequences, J. Time Ser. Anal., 42 (2021), 622–652.

  5. N. M. Bingham, Szegő’s theorem and its probabilistic descendants, Probability Surveys, 9 (2012), 287–324.

  6. P. J. Brockwell and R. A. Davis, Time Series: Theory and Methods, 2nd ed., Springer (New York, 1991).

  7. L. D. Davisson, Prediction of time series from finite past, J. Soc. Indust. Appl. Math., 13 (1965), 819–826.

  8. M. S. Ginovyan, Asymptotic behavior of the prediction error for stationary random sequences, J. Cont. Math. Anal., 34 (1999), 14–33.

  9. U. Grenander and G. Szegő, Toeplitz Forms and Their Applications, University of California Press (Berkeley, Los Angeles, 1958).

  10. W. J. Pierson, Jr., Wind generated gravity waves, in: Advances in Geophysics, Academic Press (New York, 1955), pp. 93–178.

  11. F. Pollaczek, Sur une généralisation des polynômes de Legendre, C. R. Acad. Sci. Paris, 228 (1949), 1363–1365.

  12. E. A. Rakhmanov, On the asymptotics of the ratio of orthogonal polynomials. II, Math. USSR Sb., 46 (1983), 105–117.

  13. E. A. Rakhmanov, On asymptotic properties of polynomials orthogonal on the circle with weights not satisfying Szegő’s condition, Math. USSR Sb., 58 (1987), 149–167.

  14. M. Rosenblatt, Some purely deterministic processes, J. Math. Mech., 6 (1957), 801– 810.

  15. B. Simon, Orthogonal Polynomials on the Unit Circle, AMS Coll. Publ., vol. 54, Amer. Math. Soc. (Providence, RI, 2005).

  16. G. Szegő, On certain special sets of orthogonal polynomials, Proc. Amer. Math. Soc., 1 (1950), 731–737.

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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

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