# Chains with Complete Connections and One-Dimensional Gibbs Measures

@article{Fernndez2003ChainsWC,
title={Chains with Complete Connections and One-Dimensional Gibbs Measures},
author={Roberto Fern{\'a}ndez and Gr'egory Maillard},
journal={Electronic Journal of Probability},
year={2003},
volume={9},
pages={145-176}
}
• Published 1 May 2003
• Mathematics, Computer Science
• Electronic Journal of Probability
We discuss the relationship between one-dimensional Gibbs measures and discrete-time processes (chains). We consider finite-alphabet (finite-spin) systems, possibly with a grammar (exclusion rule). We establish conditions for a stochastic process to define a Gibbs measure and vice versa. Our conditions generalize well known equivalence results between ergodic Markov chains and fields, as well as the known Gibbsian character of processes with exponential continuity rate. Our arguments are purely…

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