The Annals of Applied Probability

Weak laws of large numbers in geometric probability

Mathew D. Penrose and J. E. Yukich

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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.

Article information

Source
Ann. Appl. Probab. Volume 13, Number 1 (2003), 277-303.

Dates
First available in Project Euclid: 16 January 2003

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

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60F25: $L^p$-limit theorems

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

Citation

Penrose, Mathew D.; Yukich, J. E. Weak laws of large numbers in geometric probability. The Annals of Applied Probability 13 (2003), no. 1, 277--303. doi:10.1214/aoap/1042765669. http://projecteuclid.org/euclid.aoap/1042765669.


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  • BETHLEHEM, PENNSy LVANIA 18015 E-MAIL: joseph.yukich@lehigh.edu