随机最近邻型图的显式大数定律
Explicit laws of large numbers for random nearest-neighbour type graphs
摘要：
在Penrose & Yukich [Ann. Appl. Probab.，（2003）13，277--303]我们给出了$\R^d$中随机点集上几个最近邻型图的总幂加权长度的大数定律（在$L^p$意义下），$d\in\N$。其中一些结果是已知的;一些是新的。我们明确地给出了限制常数，在以前，他们已经在较低的一般性或根本没有评估。我们考虑的图包括k-最近邻图、Gabriel图、最小有向生成森林和在线最近邻图。
英文摘要：
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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