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2007 Gaussian Limts for Random Geometric Measures
Mathew Penrose
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Electron. J. Probab. 12: 989-1035 (2007). DOI: 10.1214/EJP.v12-429


Given $n$ independent random marked $d$-vectors $X_i$ with a common density, define the measure $\nu_n = \sum_i \xi_i $, where $\xi_i$ is a measure (not necessarily a point measure) determined by the (suitably rescaled) set of points near $X_i$. Technically, this means here that $\xi_i$ stabilizes with a suitable power-law decay of the tail of the radius of stabilization. For bounded test functions $f$ on $R^d$, we give a central limit theorem for $\nu_n(f)$, and deduce weak convergence of $\nu_n(\cdot)$, suitably scaled and centred, to a Gaussian field acting on bounded test functions. The general result is illustrated with applications to measures associated with germ-grain models, random and cooperative sequential adsorption, Voronoi tessellation and $k$-nearest neighbours graph.


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Mathew Penrose. "Gaussian Limts for Random Geometric Measures." Electron. J. Probab. 12 989 - 1035, 2007.


Accepted: 2 August 2007; Published: 2007
First available in Project Euclid: 1 June 2016

MathSciNet: MR2336596
Digital Object Identifier: 10.1214/EJP.v12-429

Primary: 60D05
Secondary: 52A22 , 60F05 , 60G57

Keywords: Random measures

Vol.12 • 2007
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