Electronic Journal of Probability

Gaussian Limts for Random Geometric Measures

Mathew Penrose

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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.

Article information

Electron. J. Probab., Volume 12 (2007), paper no. 35, 989-1035.

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

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Mathematical Reviews number (MathSciNet)

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60G57: Random measures 60F05: Central limit and other weak theorems 52A22: Random convex sets and integral geometry [See also 53C65, 60D05]

Random measures

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Penrose, Mathew. Gaussian Limts for Random Geometric Measures. Electron. J. Probab. 12 (2007), paper no. 35, 989--1035. doi:10.1214/EJP.v12-429. https://projecteuclid.org/euclid.ejp/1464818506

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