The Description of a Random Field by Means of Its Conditional Distribution

Let T denote the d-dimensional integer lattice. Let X = (X(t); t belongs to T) be a process taking values on a countable space E with probability law P. The pair (X, P) is called a random field. For each finite subset J of T, let Q sub J denote the conditional distribution of (X(t); t belongs to J) given the values of X(t), t belongs to J. The collection Q = (Q sub J; J subset T) is called the conditional distribution of (X, P). A characterization is given for the collection of all random fields with a given conditional distribution Q. (Author).