Asymptotics for the partial autocorrelation function of a stationary process
type: Item
instance of: scholarly article
title: Asymptotics for the partial autocorrelation function of a stationary process
zbMATH Open document ID: 0971.60039
DOI: 10.1007/BF02788986
author: Akihiko Inoue
published in: Journal d'Analyse Mathématique
publication date: 2001-03-19T00:00:00Z
review text: Let \(\{X_t\}\) be a real discrete purely nondeterministic weakly stationary process with vanishing mean. Let \(\gamma(n)\) and \(\alpha(n)\) be its autocovariance function and partial autocorrelation function, respectively. The main result of the paper can be formulated as follows. If \(\gamma(n)\sim n^{2d-1}\ell(n)\) as \(n\to\infty\) where \(-\infty<d<\frac 12\) and \(\ell\) is a slowly varying function at infinity, then \(|\alpha(n)|\sim\gamma(n) /\sum_{k=-n}^n\gamma(k)\). In particular, if \(0<d<\frac 12\), then \(|\alpha(n)|\sim d/n\). The derivation of these formulas is based on asymptotic analysis of the relevant expected prediction error using precise asymptotics for the sequences of MA\((\infty)\) and AR\((\infty)\) coefficients of the process. The author proves a discrete-time analogue of the Seghier-Dym theorem concerning the intersection of past and future of a process. This result is used in the proof of the main theorem.
Mathematics Subject Classification ID: 60G10
Mathematics Subject Classification ID: 62M10
zbMATH DE Number: 1546757
zbMATH Keywords: autocovariance function
zbMATH Keywords: partial autocorrelation function
zbMATH Keywords: Seghier-Dym theorem
zbMATH Keywords: reflection positivity
reviewed by: Jiří Anděl
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02788986
OpenAlex ID: W1975379694
description: scientific article; zbMATH DE number 1546757