Weak laws of large numbers in geometric probability
Mathew D. Penrose and J. E. Yukich
Source: Ann. Appl. Probab. Volume 13, Number 1
(2003), 277-303.
Abstract
Using a coupling argument, we establish a general weak law of large numbers for functionals of binomial point processes in d-dimensional space, with a limit that depends explicitly on the (possibly nonuniform) density of the point process. The general result is applied to the minimal spanning tree, the k-nearest neighbors graph, the Voronoi graph and the sphere of influence graph. Functionals of interest include total edge length with arbitrary weighting, number of vertices of specified degree and number of components. We also obtain weak laws of large numbers functionals of marked point processes, including statistics of Boolean models.
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Keywords: Weak law of large numbers; computational geometry; objective method; minimal spanning tree; nearest neighbors graph; Voronoi graph; sphere of influence graph; proximity graph; Boolean models
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Permanent link to this document: http://projecteuclid.org/euclid.aoap/1042765669
Digital Object Identifier: doi:10.1214/aoap/1042765669
Mathematical Reviews number (MathSciNet): MR1952000
Zentralblatt MATH identifier: 1029.60008
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BETHLEHEM, PENNSy LVANIA 18015 E-MAIL: joseph.yukich@lehigh.edu
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