Open Access
Translator Disclaimer
November 2001 Central Limit Theorems for Some Graphs in Computational Geometry
Mathew D. Penrose, J.E. Yukich
Ann. Appl. Probab. 11(4): 1005-1041 (November 2001). DOI: 10.1214/aoap/1015345393


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.


Download Citation

Mathew D. Penrose. J.E. Yukich. "Central Limit Theorems for Some Graphs in Computational Geometry." Ann. Appl. Probab. 11 (4) 1005 - 1041, November 2001.


Published: November 2001
First available in Project Euclid: 5 March 2002

zbMATH: 1044.60016
MathSciNet: MR1878288
Digital Object Identifier: 10.1214/aoap/1015345393

Primary: Primary 60F05
Secondary: 60D05

Keywords: $k$-nearest neighbors graph , central limit theorems , computational geometry , sphere of influence graph , Voronoi graph.

Rights: Copyright © 2001 Institute of Mathematical Statistics


Vol.11 • No. 4 • November 2001
Back to Top