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

Wade, Andrew (2007) Explicit laws of large numbers for random nearest-neighbour-type graphs. Advances in Applied Probability, 39 (2). pp. 326-342. ISSN 0001-8678

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Abstract

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 Lp sense) for the total power-weighted length of several nearest-neighbour type graphs on random point sets in Rd, 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.