Asymptotic behavior of the prediction error for stationary sequences

TL;DR

This paper investigates the asymptotic behavior of the best linear prediction error for stationary sequences, highlighting differences between deterministic and nondeterministic processes and their spectral properties.

Contribution

It provides a survey of recent results on prediction error asymptotics, especially focusing on the less-studied deterministic processes and their spectral characteristics.

Findings

Nondeterministic processes' prediction error depends on dependence structure and spectral density derivatives.

Deterministic processes' prediction error is influenced by spectral spectrum geometry and singularities.

The paper emphasizes the contrasting spectral influences on prediction errors for different process types.

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),$ $-n\le t\le-1$, as $n$ goes to infinity. This behavior depends on the regularity (deterministic or nondeterministic) and on the dependence structure of the underlying observed process $X(t)$. In this paper we consider this problem both for deterministic and nondeterministic processes and survey some recent results. We focus on the less investigated case - deterministic processes. It turns out that for nondeterministic processes the asymptotic behavior of the prediction error is determined by the dependence structure of the observed process $X(t)$ and the differential properties of its spectral density $f$, while for deterministic processes it is determined by the geometric properties of the spectrum of $X(t)$ and singularities of its spectral density $f$.