Limit Theorems for Random Fields

Abstract

Limit theorems for random fields have their specific properties caused by the parameter dimensionality. As it was mentioned in Chapter 2 one should be careful to a certain degree while introducing the weak dependence conditions for random field, since it is possible that the random fields classes determined by these conditions practically do not differ from the set of independent random variables. Another difference from the theory of limit theorems for the s.r.p. consists in the wide choice of the passage to infinite volume. We will be interested in the random fields satisfying the conditions of α- and φ-mixing in a sense of Chapter 2. As for the applications to statistical physics let’s note that the limit theorems for the random fields with φ-mixing condition turned out to be interesting, since Gibbs random fields possess this property.