The Annals of Probability

Limit Theorems for Multiply Indexed Mixing Random Variables, with Application to Gibbs Random Fields

Abstract

If $d$ is a fixed positive integer, let $\Lambda_N$ be a finite subset of $Z^d$, the lattice points of $\mathbb{R}^d$, with $\Lambda_N \uparrow Z^d$ and satisfying certain regularity properties. Let $(X_{N, Z})_{Z\in\Lambda_N}$ be a collection of random variables which satisfy a mixing condition and whose partial sums $X_N = \sum_{Z\in\Lambda_N} X_{N, Z}$ have uniformly bounded variances. Limit theorems, including a central limit theorem, are obtained for the sequence $X_N$. The results are applied to Gibbs random fields known to satisfy a sufficiently strong mixing condition.

Article information

Source
Ann. Probab., Volume 6, Number 2 (1978), 207-215.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176995568

Digital Object Identifier
doi:10.1214/aop/1176995568

Mathematical Reviews number (MathSciNet)
MR482951

Zentralblatt MATH identifier
0374.60033

JSTOR