Extensions of Rosenblatt's results on the asymptotic behavior of the prediction error for deterministic stationary sequences. (27th December 2020)

• Record Type: Journal Article
• Authors: Babayan, Nikolay M.; Ginovyan, Mamikon S.; Taqqu, Murad S.
• Other Names: Bradley Richard C. guestEditor.; Davis Richard A. guestEditor.; Politis Dimitris N. guestEditor.
• Is Part Of: Journal of time series analysis. Volume 42:Number 5/6(2021)
• Page Start: 622
• Page End: 652
• Publication Date: 2020-12-27
• Subjects: Asymptotic behavior of the prediction error -- deterministic stationary process -- Rakhmanov's theorem -- Rosenblatt's theorems -- transfinite diameter -- Verblunsky's coefficients.; Time-series analysis -- Periodicals; 519.232
• DOI: 10.1111/jtsa.12572
• Languages: English
• ISSNs: 0143-9782

Abstract

Abstract : 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 ≤ t ≤ −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. of Math. and Mech., 6(6), 801–10, 1957), 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 → ∞ . 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.