The Annals of Applied Probability

Central Limit Theorems for Some Graphs in Computational Geometry

Mathew D. Penrose and J.E. Yukich

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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.

Article information

Source
Ann. Appl. Probab., Volume 11, Number 4 (2001), 1005-1041.

Dates
First available in Project Euclid: 5 March 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1015345393

Digital Object Identifier
doi:10.1214/aoap/1015345393

Mathematical Reviews number (MathSciNet)
MR1878288

Zentralblatt MATH identifier
1044.60016

Subjects
Primary: Primary 60F05
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

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

Citation

Penrose, Mathew D.; Yukich, J.E. Central Limit Theorems for Some Graphs in Computational Geometry. Ann. Appl. Probab. 11 (2001), no. 4, 1005--1041. doi:10.1214/aoap/1015345393. https://projecteuclid.org/euclid.aoap/1015345393


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