The Annals of Probability

A Central Limit Theorem With Applications to Percolation, Epidemics and Boolean Models

Mathew D. Penrose

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Abstract

Suppose $X = (X_x)_{x\in\mathbb{Z}^d}$ is a white noise process, and $H (B)$, defined for finite subsets $B$ of $\math{Z}^d$, is determined in a stationary way by the restriction of $X$ to $B$. Using a martingale approach, we prove a central limit theorem (CLT) for $H$ as $B$ becomes large, subject to $H$ satisfying a “stabilization” condition (the effect of changing $X _x$ at a single site needs to be local). This CLT is then applied to component counts for percolation and Boolean models, to the size of the big cluster for percolation on a box, and to the final size of a spatial epidemic.

Article information

Source
Ann. Probab., Volume 29, Number 4 (2001), 1515-1546.

Dates
First available in Project Euclid: 5 March 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1015345760

Digital Object Identifier
doi:10.1214/aop/1015345760

Mathematical Reviews number (MathSciNet)
MR1880230

Zentralblatt MATH identifier
1044.60015

Subjects
Primary: 60F05: Central limit and other weak theorems 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Geometric probability percolation Boolean model central limit theorem martingale

Citation

Penrose, Mathew D. A Central Limit Theorem With Applications to Percolation, Epidemics and Boolean Models. Ann. Probab. 29 (2001), no. 4, 1515--1546. doi:10.1214/aop/1015345760. https://projecteuclid.org/euclid.aop/1015345760


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