Hilbert space representations of \(m\)-dependent processes
type: Item
instance of: scholarly article
title: Hilbert space representations of \(m\)-dependent processes
zbMATH Open document ID: 0802.60034
author: Vincent de Valk
published in: The Annals of Probability
publication date: 1994-12-15T00:00:00Z
review text: This paper deals with the investigation of the structure of \(m\)-dependent discrete time stochastic processes and can be viewed as continuation of the author's successful research work in this field. The main result of this paper consists in showing that every stationary one-dependent random sequence (with finite state space) admits a so-called Hilbert space representation (HSR). Further, if the Hilbert space which this representation is based on is two-dimensional, then the corresponding one-dependent sequence \(X_ n\), \(n\in N\), is a two-block-factor, i.e. there is a function \(f\) of two variables such that \(X_ n= f(Y_ n, Y_{n+1})\) with i.i.d. r.v.'s \(Y_ n\), \(n\in N\). As shown by \textit{J. Aaronson}, \textit{D. Gilat} and \textit{M. Keane} [J. Theor. Probab. 5, No. 3, 545-561 (1992; Zbl 0754.60070)] there exist indeed examples of one-dependent sequences which cannot be a two-block factor.
Mathematics Subject Classification ID: 60G10
Mathematics Subject Classification ID: 60B11
Mathematics Subject Classification ID: 60G05
zbMATH DE Number: 440248
zbMATH Keywords: one-dependent Markov chain
zbMATH Keywords: \(m\)-dependent discrete time stochastic processes
zbMATH Keywords: Hilbert space representation
zbMATH Keywords: two-block factor
reviewed by: Lothar Heinrich
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176989130
OpenAlex ID: W2034489958
DOI: 10.1214/AOP/1176989130
description: scientific article; zbMATH DE number 440248