Electronic Journal of Probability
- Electron. J. Probab.
- Volume 12 (2007), paper no. 35, 989-1035.
Gaussian Limts for Random Geometric Measures
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.
Electron. J. Probab., Volume 12 (2007), paper no. 35, 989-1035.
Accepted: 2 August 2007
First available in Project Euclid: 1 June 2016
Permanent link to this document
Digital Object Identifier
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]
This work is licensed under aCreative Commons Attribution 3.0 License.
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