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January 1999 A Strong Law for the Longest Edge of the Minimal Spanning Tree
Mathew D. Penrose
Ann. Probab. 27(1): 246-260 (January 1999). DOI: 10.1214/aop/1022677261

Abstract

Suppose $X_1, X_2, X_3,\ldots$ are independent random points in $\mathbf{R}^d,d\geq 2$, with common density $f$, having connected compact support $\Omega$ with smooth boundary $\partial\Omega$, with $f|_{\Omega}$ continuous. Let $M_{n}$ denote the smallest $r$ such that the union of balls of diameter $r$ centered at the first $n$ points is connected. Let $\theta$ denote the volume of the unit ball. Then as $n\to\infty$, $$n\theta M^d_n/\log n \to \max\Big(\big(\min\limits_{\Omega}f\big)^{-1},2(1 - 1/d)\big(\min\limits_{\partial\Omega}f\big)^{-1}\Big),\quad\text{a.s.}$$

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Mathew D. Penrose. "A Strong Law for the Longest Edge of the Minimal Spanning Tree." Ann. Probab. 27 (1) 246 - 260, January 1999. https://doi.org/10.1214/aop/1022677261

Information

Published: January 1999
First available in Project Euclid: 29 May 2002

zbMATH: 0944.60015
MathSciNet: MR1681114
Digital Object Identifier: 10.1214/aop/1022677261

Subjects:
Primary: 60D05
Secondary: 60F15, 60G70

Rights: Copyright © 1999 Institute of Mathematical Statistics

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Vol.27 • No. 1 • January 1999
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