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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

Abstract

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.

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Mathew D. Penrose. J.E. Yukich. "Central Limit Theorems for Some Graphs in Computational Geometry." Ann. Appl. Probab. 11 (4) 1005 - 1041, November 2001. https://doi.org/10.1214/aoap/1015345393

Information

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

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

Subjects:
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

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Vol.11 • No. 4 • November 2001
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