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Rooted edges of a minimal directed spanning tree on random points

Part of: Limit theorems Graph theory Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Z. D. Bai ,
Sungchul Lee  and
Mathew D. Penrose
Z. D. Bai*
Affiliation:
Northeast Normal University and National University of Singapore
Sungchul Lee*
Affiliation:
Yonsei University
Mathew D. Penrose*
Affiliation:
University of Bath
*
∗ Postal address: School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, P. R. China.
∗∗ Postal address: Department of Mathematics, Yonsei University, Seoul 120-749, Korea. Email address: sungchul@yonsei.ac.kr
∗∗∗ Postal address: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK.
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Abstract

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For n independent, identically distributed uniform points in [0, 1]d, d ≥ 2, let Ln be the total distance from the origin to all the minimal points under the coordinatewise partial order (this is also the total length of the rooted edges of a minimal directed spanning tree on the given random points). For d ≥ 3, we establish the asymptotics of the mean and the variance of Ln, and show that Ln satisfies a central limit theorem, unlike in the case d = 2.

Keywords

Minimal spanning treemultivariate extremecentral limit theorem

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60G70: Extreme value theory; extremal processes
Secondary: 05C80: Random graphs 60F05: Central limit and other weak theorems
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 38 , Issue 1 , March 2006 , pp. 1 - 30
Copyright
Copyright © Applied Probability Trust 2006 

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