Description of random fields by systems of conditional distributions

TL;DR

This paper investigates how to characterize lattice positive random fields through systems of finite-dimensional conditional distributions, providing necessary and sufficient conditions for their representation and discussing potential applications.

Contribution

It offers a comprehensive analysis of conditions under which systems of conditional distributions uniquely define random fields, extending known results to new classes.

Findings

Necessary and sufficient conditions for systems to be conditional distributions

Characterization of Dobrushin-type systems with only sufficient conditions

Discussion of applications of these systems in modeling

Abstract

In this paper, we consider the direct and inverse problems of the description of lattice positive random fields by various systems of finite-dimensional (as well as one-point) probability distributions parameterized by boundary conditions. In the majority of cases, we provide necessary and sufficient conditions for the system to be a conditional distribution of a (unique) random field. The exception is Dobrushin-type systems for which only sufficient conditions are known. Also, we discuss possible applications of the considered systems.