# Chains with complete connections and one-dimensinnal Gibbs measures

Abstract : We introduce an statistical mechanical formalism for the study of discrete-time stochastic processes (chains) with which we prove: (i) General properties of extremal chains, including triviality on the tail $\sigma$-algebra, short-range correlations, realization via infinite-volume limits and ergodicity. (ii) Two new sufficient conditions for the uniqueness of the consistent chain. (iii) Results on loss of memory and mixing properties for chains in the Dobrushin regime. We discuss the relationship between chains and one-dimensional Gibbs measures. We consider finite-alphabet systems, possibly with a grammar. We establish conditions for a chain to define a Gibbs measure and vice versa. We discuss the equivalence of uniqueness criteria for chains and fields and we establish bounds for the continuity rates of the respective systems of finite-volume conditional probabilities. We prove a (re)construction theorem for specifications starting from single-site conditioning.
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https://tel.archives-ouvertes.fr/tel-00005285
Contributor : Gregory Maillard <>
Submitted on : Wednesday, March 10, 2004 - 2:17:24 PM
Last modification on : Tuesday, February 5, 2019 - 11:44:10 AM
Long-term archiving on: : Wednesday, September 12, 2012 - 2:05:32 PM

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• HAL Id : tel-00005285, version 1

### Citation

Gregory Maillard. Chains with complete connections and one-dimensinnal Gibbs measures. Mathematics [math]. Université de Rouen, 2003. English. ⟨tel-00005285⟩

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