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Chains with Complete Connections: General Theory, Uniqueness, Loss of Memory and Mixing Properties

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Abstract

We introduce a statistical mechanical formalism for the study of discrete-time stochastic processes with which we prove: (i) General properties of extremal chains, including triviality on the tail σ-algebra, short-range correlations, realization via infinite-volume limits and ergodicity. (ii) Two new sufficient conditions for the uniqueness of the consistent chain. The first one is a transcription of a criterion due to Georgii for one-dimensional Gibbs measures, and the second one corresponds to the Dobrushin’s criterion in statistical mechanics. (iii) Results on loss of memory and mixing properties for chains in the Dobrushin regime. These results are complementary to those existing in the literature, and generalize the Markovian results based on the Dobrushin ergodic coefficient.

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References

  • H. Berbee (1987) ArticleTitleChains with complete connections: Uniqueness and Markov representation Prob. Th. Rel. Fields 76 243–253

    Google Scholar 

  • X. Bressaud, R. Fernández, and A. Galves. Decay of correlations for non Hölderian dynamics. a coupling approach. Elect. J. Prob. 4, 1999 ( http://www.math.washington.edu/ejpecp/).

  • N. Berger, C. Hoffman, and V. Sidoravicius. Nonuniqueness for specifications in l2+ε. Preprint arXiv:mat.PR/0312344 (2003).

  • M. Bramson S. A. Kalikow (1993) ArticleTitleNonuniqueness in g-functions Israel J. Math. 84 153–160 Occurrence Handle0786.60043

    MATH  Google Scholar 

  • R. M. Dudley. Real analysis and probability (Cambridge University Press, 2002).

  • R. Fernández and G. Maillard. Chains with complete connections and one-dimensional Gibbs measures. Elect. J. Prob., 9 (2004). (http://www.math.washington.edu/ejpecp/).

  • R. Fernández G. Maillard (2004) ArticleTitleChains and specifications Markov Proc. Rel. Fields 10 435–456

    Google Scholar 

  • H.-O. Georgii. Stochastische Felder und ihre Anwendung auf Interaktionssysteme. Lecture Notes, Institut für Angewandte Mathematik, Universität Heidelberg (1974).

  • H.-O. Georgii. Gibbs Measures and Phase Transitions. Walter de Gruyter (de Gruyter Studies in Mathematics, Vol. 9), Berlin–New York (1988).

  • T. E. Harris (1955) ArticleTitle On chains of infinite order Pacific J. Math. 5 707–724

    Google Scholar 

  • M. Iosifescu (1992) ArticleTitleA coupling method in the theory of dependence with complete connections according to Doeblin Rev. Roum. Math. Pures et Appl. 37 59–65

    Google Scholar 

  • M. Iosifescu A. Spataru (1973) ArticleTitleOn denumerable chains of infinite order. Z. Wahrscheinlichkeitstheorie verw Gebiete 27 195–214

    Google Scholar 

  • A. Johansson A. Öberg (2003) ArticleTitleSquare summability of variations of g-functions and uniqueness of g-measures Math. Res. Lett. 10 587–601

    Google Scholar 

  • S. Kalikow (1990) ArticleTitleRandom Markov processes and uniform martingales Isr. J. Math. 71 IssueID1 33–54

    Google Scholar 

  • M. Keane (1972) ArticleTitleStrongly mixing g-measures Inventiones Math. 16 309–324

    Google Scholar 

  • Y. Lacroix. A note on weak-* perturbations of g-measures. The Indian Journal of Statistics 62:331–338 (2000).

    Google Scholar 

  • S. P. Lalley (1986) ArticleTitleRegeneration representation for one-dimensional Gibbs states Ann. Prob. 14 1262–1271

    Google Scholar 

  • G. Maillard. Chaines à liaisons complètes et mesures de Gibbs unidimensionnelles. PhD thesis, Université de Rouen (2003).

  • O. Onicescu G. Mihoc (1935) ArticleTitleSur les chaines statistiques C. R. Acad. Sci. Paris 200 511–512

    Google Scholar 

  • D. Ruelle (1968) ArticleTitleStatistical mechanics of a one-dimensional lattice gas Commun. Math. Phys. 9 267–278

    Google Scholar 

  • B. Simon. The Statistical Mechanics of Lattice Gases. Series in Physics. Princeton (1993).

  • O. Stenflo. Uniqueness in g-measures. Institute of Physics {P}ublishing 16:403–410 (2002). Nonlinearity.

  • A. C. D. Enter Particlevan R. Fernández A. D. Sokal (1993) ArticleTitleRegularity properties and pathologies of position-space renormalization-group transformations: scope and limitations of Gibbsian theory J. Stat. Phys. 72 879–1167

    Google Scholar 

  • P. Walters (1975) ArticleTitleRuelle’s operator theorem and g-measures Trans. Am. Math. Soc. 214 375–387

    Google Scholar 

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Authors and Affiliations

  1. Laboratoire de Mathématiques Raphaël Salem, Site Colbert, UMR 6085 CNRS-Université de Rouen, F-76821 Mont Saint Aignan, France

    Roberto Fernández & Grégory Maillard

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  1. Roberto Fernández
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  2. Grégory Maillard
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Correspondence to Roberto Fernández.

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Fernández, R., Maillard, G. Chains with Complete Connections: General Theory, Uniqueness, Loss of Memory and Mixing Properties. J Stat Phys 118, 555–588 (2005). https://doi.org/10.1007/s10955-004-8821-5

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  • DOI: https://doi.org/10.1007/s10955-004-8821-5

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