# Une Approche vers la Description et l'Identification d'une Classe de Champs Aléatoires

Abstract : A new approach towards description of random fields on the $\nu$ -dimensional integer lattice $Z^\nu$ is presented. The random fields are described by means of some functions of subsets of $Z^\nu$ , namely $P$-functions, $Q$-functions, $H$-functions, $Q$-systems, $H$-systems and one-point systems. Interconnection with classical Gibbs description is shown. Special attention is paid to quasilocal case. Non-Gibbsian random fields are also considered. A general scheme for constructing non-Gibbsian random fields is given. The solution to Dobrushin's problem concerning the description of random field by means of its one-point conditional distributions is deduced from our approach. Further the problems of parametric estimation for Gibbs random fields is considered. The field is supposed to be specified through a translation invariant local one-point system. An estimator of one-point system is constructed as a ratio of some empirical conditional frequencies, and its uniform exponential and $L^p$ consistencies are proved. Finally the nonparametric problem of estimation of quasilocal one-point systems is considered. An estimator of one-point system is constructed by the method of sieves, and its exponential and $L^p$ consistencies are proved in different setups. The results hold regardless of non-uniqueness and translation invariance breaking.
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https://tel.archives-ouvertes.fr/tel-00748012
Contributor : Serguei Dachian <>
Submitted on : Saturday, November 3, 2012 - 2:01:21 PM
Last modification on : Friday, May 29, 2020 - 4:02:28 PM
Long-term archiving on: : Monday, February 4, 2013 - 3:41:31 AM

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• HAL Id : tel-00748012, version 1

### Citation

Serguei Dachian. Une Approche vers la Description et l'Identification d'une Classe de Champs Aléatoires. Probability [math.PR]. Université Pierre et Marie Curie - Paris VI, 1999. English. ⟨tel-00748012⟩

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