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 Teor. Veroyatnost. i Primenen., 1968, Volume 13, Issue 2, Pages 201–229 (Mi tvp839)

The description of the random field by its conditional distributions and its regularity conditions

R. L. Dobrushin

Moscow

Abstract: Suppose that for any finite set $\{t_1,…,t_n\}\subset T^\nu$, where $T^\nu$ is the $\nu$-dimensional cubic lattice, and for any $x_i$, $x(t)$, conditional probabilities
$$\mathbf P\{\xi(t_1)=x_1,…,\xi(t_n)=x_n\mid\xi(t)=x(t), t\in T^\nu, t\ne t_i, i=1,…,n\}$$
corresponding to a random field with a finite number of its values $\xi(t)$ and known and have some natural properties of consistency. The problem is to find out if it is possibjle to find absolute probabilities $\mathbf\{\xi(t_1)=x_1,…,\xi(t_n)=x_n\}$, by which the given family of conditional probabilities is generated. It is proved that there exists a solution of this, problem and that in case $\nu=1$ it is unique. For $\nu>1$, the uniqueness can be proved if conditional distributions are close in a certain sense to those of a field of independent variables. Some systems of statistical physics with phase transitions give us examples when the solution is not unique. In more detail this question is considered in [4]. We prove also that the uniqueness is equivalent to one of the forms of regularity conditions of the field.

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English version:
Theory of Probability and its Applications, 1968, 13:2, 197–224

Bibliographic databases:

Citation: R. L. Dobrushin, “The description of the random field by its conditional distributions and its regularity conditions”, Teor. Veroyatnost. i Primenen., 13:2 (1968), 201–229; Theory Probab. Appl., 13:2 (1968), 197–224

Citation in format AMSBIB
\Bibitem{Dob68} \by R.~L.~Dobrushin \paper The description of the random field by its conditional distributions and its regularity conditions \jour Teor. Veroyatnost. i Primenen. \yr 1968 \vol 13 \issue 2 \pages 201--229 \mathnet{http://mi.mathnet.ru/tvp839} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=231434} \zmath{https://zbmath.org/?q=an:0184.40403|0177.45202} \transl \jour Theory Probab. Appl. \yr 1968 \vol 13 \issue 2 \pages 197--224 \crossref{https://doi.org/10.1137/1113026} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
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