Autoregressive representations of multivariate stationary stochastic processes
type: Item
instance of: scholarly article
title: Autoregressive representations of multivariate stationary stochastic processes
zbMATH Open document ID: 0639.60045
DOI: 10.1007/BF00356109
author: M. Pourahmadi
publication date: 1988-01-01T00:00:00Z
review text: Consider a q-variate weakly stationary stochastic process \(\{X_ n\}\) with the spectral density W. The problem of autoregressive representation of \(\{X_ n\}\) or equivalently the autoregressive representation of the linear least squares predictor of \(X_ n\) based on the infinite past is studied. It is shown that for every W in a large class of densities, the corresponding process has a mean convergent autoregressive representation. This class includes as special subclasses, the densities studied by \textit{P. Masani} [Acta Math. 104, 141-162 (1960; Zbl 0096.115)] and the author [J. Multivariate Anal. 16, 265-536 (1985; Zbl 0583.62086)]. As a consequence it is shown that the condition \(W^{-1}\in L\) \(1_{q\times q}\) or minimality of \(\{X_ n\}\) is dispensable for this problem. When W is not in this class or when W has zeros of order 2 or more, it is shown that \(\{X_ n\}\) has a mean Abel summable or mean compounded Cesáro summable autoregressive representation.
Mathematics Subject Classification ID: 60G12
Mathematics Subject Classification ID: 60G10
Mathematics Subject Classification ID: 62M10
Mathematics Subject Classification ID: 60G25
zbMATH DE Number: 4042960
zbMATH Keywords: weakly stationary stochastic process
zbMATH Keywords: autoregressive representation
zbMATH Keywords: linear least squares predictor
zbMATH Keywords: Cesáro summable
zbMATH Keywords: Abel summable
MaRDI profile type: Publication
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 4042960