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

Wade, Andrew R.

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Abstract

Under the unifying umbrella of a general result of Penrose and Yukich (Annals of Applied Probability 13 (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.

Citation

Wade, A. R. (2007). Explicit laws of large numbers for random nearest-neighbour-type graphs. Advances in Applied Probability, 39(2), 326-342. https://doi.org/10.1239/aap/1183667613

Journal Article Type Article
Publication Date Jun 1, 2007
Deposit Date Oct 4, 2012
Publicly Available Date Mar 29, 2024
Journal Advances in Applied Probability
Print ISSN 0001-8678
Electronic ISSN 1475-6064
Publisher Applied Probability Trust
Peer Reviewed Peer Reviewed
Volume 39
Issue 2
Pages 326-342
DOI https://doi.org/10.1239/aap/1183667613
Keywords Nearest-neighbour-type graph, Law of large numbers, Spanning forest, Spatial network evolution.

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