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

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 σ n 2 ( f ) in predicting X ( 0 ) given X ( t ) , - n ≤ t ≤ - 1 , as n goes to infinity. We consider a c...

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Bibliographic Details
Published in:Acta mathematica Hungarica 2022-08, Vol.167 (2), p.501-528
Main Authors: BABAYAN, N., GINOVYAN, M.
Format: Article
Language:English
Subjects:
Asymptotic properties
Density
Mathematics
Mathematics and Statistics
Predictions
Stationary processes
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Summary: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 σ n 2 ( f ) in predicting X ( 0 ) given X ( t ) , - n ≤ t ≤ - 1 , as n goes to infinity. We consider a class of spectral densities of the form f = f d g , 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 σ n 2 ( f ) behaves like a power as n → ∞ . Examples illustrate the obtained results.
ISSN:0236-5294
1588-2632
DOI:10.1007/s10474-022-01248-9