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.
Publié le : 2003-01-14
Classification:
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,
60D05,
60F25
@article{1042765669,
author = {Penrose, Mathew D. and Yukich, J. E.},
title = {Weak laws of large numbers in geometric probability},
journal = {Ann. Appl. Probab.},
volume = {13},
number = {1},
year = {2003},
pages = { 277-303},
language = {en},
url = {http://dml.mathdoc.fr/item/1042765669}
}
Penrose, Mathew D.; Yukich, J. E. Weak laws of large numbers in geometric probability. Ann. Appl. Probab., Tome 13 (2003) no. 1, pp. 277-303. http://gdmltest.u-ga.fr/item/1042765669/