Electronic Journal of Probability

Gaussian Limts for Random Geometric Measures

Mathew Penrose

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464818506

Digital Object Identifier
doi:10.1214/EJP.v12-429

Mathematical Reviews number (MathSciNet)
MR2336596

Subjects
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]

Keywords
Random measures

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

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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