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A central limit theorem with applications to percolation, epidemics and Boolean models. (English) Zbl 1044.60015

This paper provides a central limit theorem (CLT) for a large class of stationary, set-indexed functionals \(H(X, B_n)\) defined on a family \(X= (X_z,\,z\in\mathbb{Z}^d)\) of independent identically distributed random elements taking values in some measurable space \((E,{\mathcal E})\), where \(B_n\) runs through an increasing space-filling family of sets. A stability condition is imposed on \(H(\cdot)\) in terms of the sequence increments \(\Delta_0(B_n)= H(X, B_n)- H(X',B_n)\), where \(X'\) is the process \(X\) with \(X_0\) replaced by an independent copy \(X^*\) of \(X_0\). It is assumed that \(\Delta_0(B_n)\) converges in probability to some random variable \(\Delta_0(\infty)\). Under some regularity and moment conditions the asymptotic normality of the sequence \(| B_n|^{-1/2}(H(X, B_n)- EH(X, B_n))\) is proved by employing a CLT for martingal difference arrays. This CLT is applied to component counts for percolation and Boolean models, to the size of the biggest cluster for percolation on a box, and to the final size of a spatial epidemic.

MSC:

60F05 Central limit and other weak theorems
60D05 Geometric probability and stochastic geometry
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
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