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Attractive regular stochastic chains: perfect simulation and phase transition

Published online by Cambridge University Press:  03 April 2013

SANDRO GALLO
Affiliation:
Departamento de Métodos Estatísticos, Instituto de Matemática, Universidade Federal de Rio de Janeiro, Caixa Postal 68350, CEP 21941-909, Rio de Janeiro, Brasil email sandro@im.ufrj.br
DANIEL Y. TAKAHASHI
Affiliation:
Neuroscience Institute and Psychology Department, Princeton University, Greenhall, Princeton, NJ 08540, USA email takahashiyd@gmail.com

Abstract

We prove that uniqueness of the stationary chain, or equivalently, of the $g$-measure, compatible with an attractive regular probability kernel is equivalent to either one of the following two assertions for this chain: (1) it is a finitary coding of an independent and identically distributed (i.i.d.) process with countable alphabet; (2) the concentration of measure holds at exponential rate. We show in particular that if a stationary chain is uniquely defined by a kernel that is continuous and attractive, then this chain can be sampled using a coupling-from-the-past algorithm. For the original Bramson–Kalikow model we further prove that there exists a unique compatible chain if and only if the chain is a finitary coding of a finite alphabet i.i.d. process. Finally, we obtain some partial results on conditions for phase transition for general chains of infinite order.

Type
Research Article
Copyright
Copyright ©2013 Cambridge University Press 

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