Prediction error for continuous-time stationary processes with singular spectral densities
type: Item
instance of: scholarly article
title: Prediction error for continuous-time stationary processes with singular spectral densities
zbMATH Open document ID: 1213.62146
author: M. S. Ginovyan
author: Levon V. Mikaelyan
published in: Acta Applicandae Mathematicae
publication date: 2010-05-26T00:00:00Z
review text: This paper deals with the mean square linear prediction problem for some classes of continuous-time stationary Gaussian processes. The spectral density, \(f(\lambda)\), of the underlying process is allowed to possess singularities. The authors define a relative prediction error, \(\delta(f;T)\), of a future value of the process using the past of length \(T\), compared with the whole past. They are interested in estimating the rate of decrease of \(\delta(f;T)\) to zero as \(T \rightarrow \infty\), under the assumption that the underlying process is nondeterministic and ``close'' to white noise. They obtain explicit expressions and asymptotic formulas for \(\delta(f;T)\) in the case where the spectral density possesses either zeros (the underlying model is an antipersistent process), or poles (the model is a long memory process). The results show that the asymptotic relation \(\delta(f;T) \sim 1/T\) as \(T \rightarrow \infty\) is valid whenever the spectral density \(f(\lambda)\) of the underlying process possesses at least one singularity (zero or pole) of power type. The approach to the problem, applied by the authors, is based on the Krein's theory of continual analogs of orthogonal polynomials on the unit circle and the continual analogs of Szegö's theorem on Toeplitz determinants [see, e.g., \textit{A. Böttcher} and \textit{B. Silbermann}, Analysis of Toeplitz operators. Berlin etc.: Springer (1990; Zbl 0732.47029)].
reviewed by: Ryszard Doman
Mathematics Subject Classification ID: 62M20
Mathematics Subject Classification ID: 60G25
Mathematics Subject Classification ID: 47N30
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 62M15
zbMATH DE Number: 5712234
zbMATH Keywords: stationary Gaussian process
zbMATH Keywords: prediction error
zbMATH Keywords: parameter function
zbMATH Keywords: Wiener-Hopf truncated operator
zbMATH Keywords: Szegö theorem
MaRDI profile type: Publication
full work available at URL: s10440-008-9414-0
OpenAlex ID: W2038728989
cites work: Q5535273
cites work: Asymptotic behavior of difference between a finite predictor and an infinite predictor for a weakly stationary stochastic process
cites work: Approximation of the finite prediction for a weakly stationary process
cites work: Q3330351
cites work: WIENER-HOPE Determinants with Rational Symbols
cites work: Q4040771
cites work: A problem in trigonometric approximation theory
cites work: Q4092717
cites work: Asymptotic behavior of the prediction error for stationary random sequences
cites work: Inversion of Wiener-Hopf truncated operators and prediction error for continuous time ARMA processes
cites work: Q3247378
cites work: Prediction from part of the past of a stationary process
cites work: The Asymptotic Behavior of the Prediction Error of a Stationary Sequence with a Spectral Density of Special Type
cites work: On the asymptotic behavior of the prediction error of a stationary process
cites work: Asymptotics for prediction errors of stationary processes with reflection positivity
cites work: Q4720192
cites work: Foundations of time series analysis and prediction theory
cites work: On the Extrapolation of Generalized Stationary Random Processes
cites work: Q5531487
cites work: Spectral theory of a class of canonical differential systems
cites work: Prediction d'un processus stationnaire du second ordre de covariance connue sur un intervalle fini
cites work: Q4129261
cites work: Q3882318
cites work: Q3884892
cites work: Toeplitz Determinants with Singular Generating Functions
DOI: 10.1007/S10440-008-9414-0
Recommended article: On asymptotic behavior of the prediction error for a class of deterministic stationary sequences
Recommended article: Asymptotic behavior of the prediction error for stationary random sequences
Recommended article: Asymptotic behavior of the prediction error for stationary sequences
Recommended article: Extensions of Rosenblatt's results on the asymptotic behavior of the prediction error for deterministic stationary sequences
Recommended article: Q3330351
Recommended article: Performance of discrete-time predictors of continuous-time stationary processes
Recommended article: Spectral characterization of the Wold–Zasuhin decomposition and prediction-error operator
Recommended article: Q3755336
Recommended article: Q3715960
Recommended article: Q3717923
description: scientific article; zbMATH DE number 5712234