Une Approche Vers La Description Et L'identification D'une Classe De Champs Aleatoires.
Sergueï Dachian · HAL (Le Centre pour la Communication Scientifique Directe) · 1999
A new approach towards description of random fields on the $ u$ -dimensional integer lattice $Z^ u$ is presented. The random fields are described by means of some functions of subsets of $Z^ u$ , 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.