Central Limit Theorems for Some Graphs in Computational Geometry

Let Bn be an increasing sequence of regions in d-dimensional space with volume n and with union d. We prove a general central limit theorem for functionals of point sets, obtained either by restricting a homogeneous Poisson process to Bn, or by by taking n uniformly distributed points in Bn. The sets Bn could be all cubes but a more general class of regions Bn is considered. Using this general result we obtain central limit theorems for specific functionals such as total edge length and number of components, defined in terms of graphs such as the k-nearest neighbors graph, the sphere of influence graph and the Voronoi graph.