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January 2003 Weak laws of large numbers in geometric probability
Mathew D. Penrose, J. E. Yukich
Ann. Appl. Probab. 13(1): 277-303 (January 2003). DOI: 10.1214/aoap/1042765669


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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Mathew D. Penrose. J. E. Yukich. "Weak laws of large numbers in geometric probability." Ann. Appl. Probab. 13 (1) 277 - 303, January 2003.


Published: January 2003
First available in Project Euclid: 16 January 2003

zbMATH: 1029.60008
MathSciNet: MR1952000
Digital Object Identifier: 10.1214/aoap/1042765669

Primary: 60D05
Secondary: 60F25

Keywords: Boolean models , computational geometry , Minimal spanning tree , nearest neighbors graph , objective method , proximity graph , sphere of influence graph , Voronoi graph , Weak law of large numbers

Rights: Copyright © 2003 Institute of Mathematical Statistics


Vol.13 • No. 1 • January 2003
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