The structure of the Naimark dilation and Gaussian stationary processes
type: Item
instance of: scholarly article
title: The structure of the Naimark dilation and Gaussian stationary processes
zbMATH Open document ID: 0562.47005
DOI: 10.1007/BF01202813
published in: Integral Equations and Operator Theory
publication date: 1985-01-01T00:00:00Z
review text: The main concern of this paper is the study of positive definite block Toeplitz forms over \({\mathbb{N}}\). In the paper ''On the structure of Naimark dilation'' by \textit{T. Constantinescu} which appeared in J. Oper. Theory 12, 159-175 (1984), the analysis of the structure of the Naimark dilation of such a form is done using so-called ''choice sequences'' (a generalization to the operator case of the classical notion of ''Schur sequence''). This paper uses this analysis in order to compute some angles in Gaussian stationary processes. This is possible because such a process is completely defined by its correlation matrix, which is a positive definite Toeplitz form. The fact that choice sequences are useful for angle computations is shown in Section 4. In Section 5, this is applied for obtaining a criterium for informational regularity of the process in terms of the associated choice sequence. More explicit results are obtained for the scalar case in Section 6, where the connections with the entropy and Szegö limit theorems are also presented. Recently, these results were extended to the non-Toeplitz case.
Mathematics Subject Classification ID: 47A20
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60G25
zbMATH DE Number: 3895689
zbMATH Keywords: positive definite block Toeplitz forms
zbMATH Keywords: structure of the Naimark dilation
zbMATH Keywords: choice sequences
zbMATH Keywords: Schur sequence
zbMATH Keywords: Gaussian stationary processes
zbMATH Keywords: correlation matrix
zbMATH Keywords: angle computations
zbMATH Keywords: entropy
zbMATH Keywords: Szegö limit theorems
author: Grigore Arsene
author: T. Constantinescu
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01202813
OpenAlex ID: W2038565141
description: scientific article; zbMATH DE number 3895689