## 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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### References:

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