Local Limit Theorems for Sums of Finite Range Potentials of a Gibbsian Random Field

Abstract

Local limit theorems are derived for sums of finite range $\mathbb{Z}$-valued potential functions of an iid random field. The resulting approximations turn out to be mixtures of standard normal densities for lattice distributions supported by residue classes of integers. The mixing weights are equal to the probability that the sum of potential functions lies in such a residue class and are nonasymptotic and computable. For finite range potential functions of a stationary Gibbsian random field with bounded and finite range interactions, conditions are given under which the global central limit theorem implies the classical local limit theorem.