## The Annals of Probability

### 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.

#### Article information

Source
Ann. Probab., Volume 18, Number 2 (1990), 810-828.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176990859

Digital Object Identifier
doi:10.1214/aop/1176990859

Mathematical Reviews number (MathSciNet)
MR1055434

Zentralblatt MATH identifier
0704.60019

JSTOR