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Explicit laws of large numbers for random nearest-neighbour-type graphs

Part of: Geometric probability and stochastic geometry Limit theorems

Published online by Cambridge University Press:  01 July 2016

Andrew R. Wade
Andrew R. Wade*
Affiliation:
University of Bristol
*
∗ Postal address: Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK. Email address: andrew.wade@bristol.ac.uk
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Abstract

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Under the unifying umbrella of a general result of Penrose and Yukich (Annals of Applied Probability13 (2003), 277-303) we give laws of large numbers (in the Lp sense) for the total power-weighted length of several nearest-neighbour-type graphs on random point sets in ℝd, d ∈ ℕ. Some of these results are known; some are new. We give limiting constants explicitly, where previously they have been evaluated in less generality or not at all. The graphs we consider include the k-nearest-neighbours graph, the Gabriel graph, the minimal directed spanning forest, and the on-line nearest-neighbour graph.

Keywords

Nearest-neighbour-type graphlaw of large numbersspanning forestspatial network evolution

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60F25: $L^p$-limit theorems
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 39 , Issue 2 , June 2007 , pp. 326 - 342
Copyright
Copyright © Applied Probability Trust 2007 

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