Let $(B_n)$ be an increasing sequence of regions in $d$ -dimensional
space with volume $n$ and with union $\mathbb{R}^d$. We prove a general central
limit theorem for functionals of point sets, obtained either by restricting a
homogeneous Poisson process to $(B_n)$, or by by taking $n$ uniformly
distributed points in $(B_n)$. The sets $(B_n)$ could be all cubes but a more
general class of regions$(B_n)$ is considered. Using this general result we
obtain central limit theorems for specific functionals suchas total edge
lengthand number of components, defined in terms of graphs such as the
$k$-nearest neighbors graph, the sphere of influence graph and the Voronoi
graph.
Publié le : 2001-11-14
Classification:
Central limit theorems,
computational geometry,
$k$-nearest neighbors graph,
sphere of influence graph,
Voronoi graph.,
Primary 60F05,
60D05
@article{1015345393,
author = {Penrose, Mathew D. and Yukich, J.E.},
title = {Central Limit Theorems for Some Graphs in Computational
Geometry},
journal = {Ann. Appl. Probab.},
volume = {11},
number = {2},
year = {2001},
pages = { 1005-1041},
language = {en},
url = {http://dml.mathdoc.fr/item/1015345393}
}
Penrose, Mathew D.; Yukich, J.E. Central Limit Theorems for Some Graphs in Computational
Geometry. Ann. Appl. Probab., Tome 11 (2001) no. 2, pp. 1005-1041. http://gdmltest.u-ga.fr/item/1015345393/