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Extremes for the minimal spanning tree on normally distributed points

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

Published online by Cambridge University Press:  01 July 2016

Mathew D. Penrose
Mathew D. Penrose*
Affiliation:
Unversity of Durham
*
∗ Postal address: Department of Mathematical Sciences, University of Durham, South Road, Durham DH1 3LE. Email address: [email protected]
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Abstract

Let n points be placed independently in ν-dimensional space according to the standard ν-dimensional normal distribution. Let Mn be the longest edge-length of the minimal spanning tree on these points; equivalently let Mn be the infimum of those r such that the union of balls of radius r/2 centred at the points is connected. We show that the distribution of (2 log n)1/2Mn - bn converges weakly to the Gumbel (double exponential) distribution, where bn are explicit constants with bn ~ (ν - 1)log log n. We also show the same result holds if Mn is the longest edge-length for the nearest neighbour graph on the points.

Keywords

Geometric probabilityminimal spanning treenearest neighbour graphextreme valuesPoisson processChen-Stein method

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60G70: Extreme value theory; extremal processes
Secondary: 05C05: Trees 90C27: Combinatorial optimization
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 30 , Issue 3 , September 1998 , pp. 628 - 639
Copyright
Copyright © Applied Probability Trust 1998 

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