Strong stationarity and de Finetti's theorem
type: Item
instance of: scholarly article
title: Strong stationarity and de Finetti's theorem
zbMATH Open document ID: 0901.28011
DOI: 10.1007/BF02788136
author: E. Jenvey
published in: Journal d'Analyse Mathématique
publication date: 1998-11-16T00:00:00Z
review text: The classical result of \textit{B. de Finetti} [Ann. Inst. Henri Poincaré 7, 1-68 (1937; Zbl 0017.07602)] states that any exchangeable process must be a mixture of independent and i.i.d. processes; if such a process is ergodic it must therefore be i.i.d. In this paper a notion weaker than exchangeability is introduced. A stationary process \(\{X_i\}\) is called strongly stationary if restricting the index \(i\) to any arithmetic progression \(\{X_{ri+b}\}\) gives a process with the same distribution. The main result is that any ergodic strongly stationary process is i.i.d. The methods used come from ergodic theory (analysing the eigenvalues of a measure-preserving dynamical system generating the process) and from the notion of IP-systems in the work of \textit{H. Furstenberg} and \textit{Y. Katznelson} [J. Anal. Math. 45, 117-168 (1985; Zbl 0605.28012)].
reviewed by: Thomas B. Ward
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 60G05
zbMATH DE Number: 1146559
zbMATH Keywords: exchangeability
zbMATH Keywords: strong stationary processes
zbMATH Keywords: eigenvalues
zbMATH Keywords: measure-preserving dynamical system
MaRDI profile type: Publication
cites work: Q5766989
cites work: Q3909268
cites work: An ergodic Szemerédi theorem for IP-systems and combinatorial theory
full work available at URL: https://doi.org/10.1007/bf02788136
OpenAlex ID: W1998497911
description: scientific article; zbMATH DE number 1146559