Almost sure convergence of multiparameter martingales for Markov random fields
type: Item
instance of: scholarly article
title: Almost sure convergence of multiparameter martingales for Markov random fields
zbMATH Open document ID: 0538.60043
author: Hans Föllmer
published in: The Annals of Probability
publication date: 1984-01-01T00:00:00Z
review text: A sufficient condition for almost sure (a.s.) convergence of bounded two- parameter martingales is the conditional independence of the underlying filtration. The author proves a.s. convergence under a weaker condition in the case where the filtration is generated by a Markov random field. He shows that bounded two-parameter martingales converge a.s. if the interaction matrix of the conditional probabilities associated with the Markov random field is bounded in the sense of Dobrushin's uniqueness condition. (This condition implies that the field is uniquely determined by its conditional probabilities). Moreover, the author constructs a Markov random field which admits no phase transition, but does admit a bounded martingale which fails to converge a.s.
reviewed by: M.Dozzi
Mathematics Subject Classification ID: 60G42
Mathematics Subject Classification ID: 60G60
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B20
zbMATH DE Number: 3856117
zbMATH Keywords: two-parameter martingales
zbMATH Keywords: Markov random field
zbMATH Keywords: Dobrushin's uniqueness condition
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176993378
OpenAlex ID: W2088323678
DOI: 10.1214/AOP/1176993378
description: scientific article; zbMATH DE number 3856117