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May, 1984 Central Limit Theorems for Associated Random Variables and the Percolation Model
J. Theodore Cox, Geoffrey Grimmett
Ann. Probab. 12(2): 514-528 (May, 1984). DOI: 10.1214/aop/1176993303

Abstract

We prove a central limit theorem for families $\{X_n(N): \mathbf{n} \in \Lambda(N)\}$ of associated random variables indexed by subsets $\Lambda(N)$ of $\mathbb{Z}^d$, as $N \rightarrow \infty$; this is an extension of the Newman-Wright invariance principle for associated stationary sequences $\{X_n: n \geq 1\}$ satisfying $\sum_n \operatorname{cov}(X_1, X_n) < \infty$, but with the stationarity property replaced by conditions on the moments of the $X$'s. The theorem has applications to the voter model and the percolation model. In the latter case, it provides an extension of a central limit theorem of the authors [4], by reducing the severity of the moment conditions. Also, we prove a central limit theorem for certain non-stationary non-associated families of random variables which arise in percolation theory. This includes, for example, a central limit theorem for the number of open clusters contained within the circuit $\gamma(n)$ of $\mathbb{Z}^2$, where $\{\gamma(n)\}$ is a sequence of circuits which satisfy a regularity condition and whose interiors $\{\overset{\circ}\gamma(n)\}$ satisfy $|\overset{\circ}\gamma(n)| \rightarrow \infty$ as $n \rightarrow \infty$.

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J. Theodore Cox. Geoffrey Grimmett. "Central Limit Theorems for Associated Random Variables and the Percolation Model." Ann. Probab. 12 (2) 514 - 528, May, 1984. https://doi.org/10.1214/aop/1176993303

Information

Published: May, 1984
First available in Project Euclid: 19 April 2007

zbMATH: 0536.60094
MathSciNet: MR735851
Digital Object Identifier: 10.1214/aop/1176993303

Subjects:
Primary: 60K35
Secondary: 60F05 , 60K99

Keywords: associated random variables , central limit theorem , percolation model , voter model

Rights: Copyright © 1984 Institute of Mathematical Statistics

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Vol.12 • No. 2 • May, 1984
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