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

Abstract

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 $f=f_{d}g$, 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.

arXiv admin note: text overlap with arXiv:2006.00430