On asymptotic behavior of the prediction error for a class of deterministic stationary sequences

• Published in: arXiv.org 2021-11
• Main Authors: Babayan, Nikolay M, Ginovyan, Mamikon S
• Format: Article
• Language: English
• Subjects: Asymptotic properties; Density; Errors; Polynomials; Predictions; Singularities; Spectra; Stationary processes
• ISSN: 2331-8422

One of the main problem in prediction theory of stationary processes \(X(t)\) is to describe the asymptotic behavior of the best linear mean squared prediction error in predicting \(X(0)\) given \( X(t),\) \(-n\le t\le-1\), as \(n\) goes to infinity. This behavior depends on the regularity (deterministic or non-deterministic) of the process \(X(t)\). In his seminal paper {\it 'Some purely deterministic processes' (J. of Math. and Mech.,} {\bf 6}(6), 801-810, 1957), for a specific spectral density that has a very high order contact with zero M. Rosenblatt showed that the prediction error behaves like a power as \(n\to\f\). In the paper Babayan et al. {\it 'Extensions of Rosenblatt's results on the asymptotic behavior of the prediction error for deterministic stationary sequences' (J. Time Ser. Anal.} {\bf 42}, 622-652, 2021), Rosenblatt's result was extended to the 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, while \(g\) is a function that can have polynomial type singularities. In this paper, we describe new extensions of the above quoted results in the case where the function \(g\) can have {\it arbitrary power type singularities}. Examples illustrate the obtained results.