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 Electronic Journal of Probability > Vol. 12 (2007) > Paper 35 open journal systems 


Gaussian limts for random geometric measures

Mathew D. Penrose, University of Bath


Abstract
Abstract. Given n independent random marked d-vectors (points) X(i) with a common density, define the random measure m as a sum of contributions from them, where the contribution from the i-th point is a (not necessarily point) measure determined by the (suitably rescaled) set of points near X(i). Technically, this means here that each contribution stabilizes with a suitable power-law decay of the tail of the radius of stabilization. For bounded test functions f on d-space, we give a central limit theorem for m(f), and deduce weak convergence of m, 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. ~


Full text: PDF

Pages: 989-1035

Published on: August 2, 2007
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Electronic Journal of Probability. ISSN: 1083-6489