Extensions of Rosenblatt's results on the asymptotic behavior of the prediction error for deterministic stationary sequences

TL;DR

This paper extends Rosenblatt's classical results on the asymptotic behavior of prediction errors for deterministic stationary sequences to broader spectral density classes, providing new insights and examples.

Contribution

It introduces a new approach to generalize Rosenblatt's findings to wider spectral density classes in prediction theory.

Findings

Prediction error behaves exponentially for spectral densities vanishing on an interval.

Prediction error behaves hyperbolically for spectral densities with high order contact with zero.

Examples illustrate the extended results and their implications.

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 non-deterministic) of the process $X(t)$. In his seminal paper {\it "Some purely deterministic processes" (J. of Math. and Mech.,} 6(6), 801-810, 1957), M. Rosenblatt has described the asymptotic behavior of the prediction error for discrete-time deterministic processes in the following two cases: (a) the spectral density $f(\lambda)$ of $X(t)$ is continuous and vanishes on an interval, (b) the spectral density $f(\lambda)$ has a very high order contact with zero. He showed that in the case (a) the prediction error variance behaves exponentially, while in the case (b), it behaves hyperbolically as $n\to\infty$. In this paper, using a new approach, we describe extensions of Rosenblatt's results to broader classes of spectral densities. Examples illustrate the obtained results.