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

TL;DR

This paper investigates the asymptotic behavior of the prediction error for deterministic stationary sequences, extending previous results to cases where spectral densities have arbitrary power-type singularities, with implications for prediction theory.

Contribution

It extends Rosenblatt's and Babayan et al.'s results to spectral densities with arbitrary power-type singularities, broadening understanding of prediction error asymptotics.

Findings

Derived new asymptotic formulas for prediction errors with arbitrary singularities.

Extended the class of spectral densities for which asymptotic behavior is characterized.

Provided examples illustrating the theoretical results.

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 $f=f_dg$, 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.