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

Wade, Andrew R.

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: Feb 13, 2013
• 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.
• Public URL: https://durham-repository.worktribe.com/output/1502824