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

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

2006-03-23

arxiv.org/abs/math/0603559v2

Advances in Applied Probability, Vol. 39 (2007), no. 2, p. 326-342

Mathematics

Probability

18 pages, 2 figures; revised presentation

Scientific paper

10.1239/aap/1183667613

Under the unifying umbrella of a general result of Penrose & Yukich [Ann. Appl. Probab., (2003) 13, 277--303] we give laws of large numbers (in the $L^p$ sense) for the total power-weighted length of several nearest-neighbour type graphs on random point sets in $\R^d$, $d\in\N$. 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.

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