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Almost sure convergence of multiparameter martingales for Markov random fields. (English) Zbl 0538.60043

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.
Reviewer: M.Dozzi

MSC:

60G42 Martingales with discrete parameter
60G60 Random fields
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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