Description of Random Fields by Means of Conditional Probabilities

Abstract

The way of determining a random process by means of its conditional probabilities has been used in probability theory quite for a long time, for example, the definition of Markov chain with the help of its transition matrix. However, it was only in connection with the demands of statistical physics, in particular, with those of the strict definition of Gibbs random field the wide possibilities of this approach were cleared up. Below we will describe some facts of this theory using the results of fundamental works /29/, /32/. As it turned out in contrast to defining a random field by means of its finite-dimensional distributions the uniqueness of a random field with the given conditional distribution does not come from its existence here. As it will be shown in this Chapter for the uniqueness to hold here it is necessary to require that the field should possess some properties of decay of correlations, for example, those of mixing. In doing this it is possible to indicate the estimates of the mixing coefficient in terms of conditional probabilities. At the end of this chapter c.l.th. will be presented for the random field under some requirements for its conditional distribution.