The Annals of Applied Probability
- Ann. Appl. Probab.
- Volume 13, Number 1 (2003), 277-303.
Weak laws of large numbers in geometric probability
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.
Ann. Appl. Probab. Volume 13, Number 1 (2003), 277-303.
First available in Project Euclid: 16 January 2003
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60F25: $L^p$-limit theorems
Penrose, Mathew D.; Yukich, J. E. Weak laws of large numbers in geometric probability. Ann. Appl. Probab. 13 (2003), no. 1, 277--303. doi:10.1214/aoap/1042765669. http://projecteuclid.org/euclid.aoap/1042765669.