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November, 1993 On Central Limit Theorems in Geometrical Probability
Florin Avram, Dimitris Bertsimas
Ann. Appl. Probab. 3(4): 1033-1046 (November, 1993). DOI: 10.1214/aoap/1177005271


We prove central limit theorems and establish rates of convergence for the following problems in geometrical probability when points are generated in the $\lbrack 0,1\rbrack^2$ cube according to a Poisson point process with parameter $n$: 1. The length of the nearest graph $N_{k,n}$, in which each point is connected to its $k$th nearest neighbor. 2. The length of the Delaunay triangulation $\operatorname{Del}_n$ of the points. 3. The length of the Voronoi diagram $\operatorname{Vor}_n$ of the points. Using the technique of dependency graphs of Baldi and Rinott, we show that the dependence range in all these problems converges quickly to 0 with high probability. Our approach also establishes rates of convergence for the number of points in the convex hull and the area outside the convex hull for points generated according to a Poisson point process in a circle.


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Florin Avram. Dimitris Bertsimas. "On Central Limit Theorems in Geometrical Probability." Ann. Appl. Probab. 3 (4) 1033 - 1046, November, 1993.


Published: November, 1993
First available in Project Euclid: 19 April 2007

zbMATH: 0784.60015
MathSciNet: MR1241033
Digital Object Identifier: 10.1214/aoap/1177005271

Primary: 60D05
Secondary: 90C27

Keywords: central limit theorems , Geometrical probability , rates of convergence

Rights: Copyright © 1993 Institute of Mathematical Statistics


Vol.3 • No. 4 • November, 1993
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