The Annals of Applied Probability

Weak laws of large numbers in geometric probability

Mathew D. Penrose and J. E. Yukich

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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

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

First available: 16 January 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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