Let B n 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 B n , or by by taking n uniformly distributed points in B n . The sets B n could be all cubes but a more general class of regions B n 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.1. Introduction. The purpose of this paper is to develop a general methodology to establish central limit theorems (CLTs) for functionals of graphs in computational geometry. Functionals of interest include total edge length, total number of edges, total number of components and total number of vertices of fixed degree. Graphs of interest include the k-nearest neighbors graph, the Voronoi and Delaunay tessellations, the sphere of influence graph, the Gabriel graph and the relative neighbor graph. These graphs are formally defined later on. In each case, the graph or its dual graph (as with the Voronoi graph) is constructed as follows: given a finite vertex set in d d ≥ 1, undirected edges are drawn from each vertex to various nearby vertices, the choice of edges to include being determined by the local point configuration according to some specified rule. Sometimes such graphs are called proximity graphs; see [3] for a precise definition.Our graphs are random in the sense that the vertex set is a random point set in d d ≥ 1. We establish CLTs for two related types of random point sets: the homogeneous Poisson point process on a large region or "window" of d and the point set consisting of a large independent sample of nonrandom sample size from the uniform distribution on such a region. By scaling, these often yield a CLT for Poisson processes of high intensity on a fixed set such as the unit cube 0 1 d , or for large independent samples of nonrandom size from the uniform distribution on a fixed set. As a by-product, we also prove the convergence of the (scaled) variance of our functionals of interest.One of our more interesting new results is a CLT for the total number of components of the k-nearest neighbors graph, either on a Poisson process or on a sample of nonrandom size, and likewise for the sphere of influence graph.

Using a coupling argument, we establish a general weak law of large numbers for functionals of binomial point processes in d-dimensional space, with a limit that depends explicitly on the (possibly non-uniform) density of the point process. The general result is applied to the minimal spanning tree, the k-nearest neighbors graph, the Voronoi graph, and the sphere of influence graph. Functionals of interest include total edge length with arbitrary weighting, number of vertices of specifed degree, and number of components. We also obtain weak laws for functionals of marked point processes, including statistics of Boolean models.

We establish Gaussian limits for general measures induced by binomial and Poisson point processes in d-dimensional space. The limiting Gaussian field has a covariance functional which depends on the density of the point process. The general results are used to deduce central limit theorems for measures induced by random graphs (nearest neighbor, Voronoi and sphere of influence graph), random sequential packing models (ballistic deposition and spatial birth-growth models) and statistics of germ-grain models.Comment: Published at http://dx.doi.org/10.1214/105051604000000594 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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We use Stein's method to obtain bounds on the rate of convergence for a class of statistics in geometric probability obtained as a sum of contributions from Poisson points which are exponentially stabilizing, i.e. locally determined in a certain sense. Examples include statistics such as total edge length and total number of edges of graphs in computational geometry and the total number of particles accepted in random sequential packing models. These rates also apply to the 1-dimensional marginals of the random measures associated with these statistics.

We establish presumably optimal rates of normal convergence with respect to the Kolmogorov distance for a large class of geometric functionals of marked Poisson and binomial point processes on general metric spaces. The rates are valid whenever the geometric functional is expressible as a sum of exponentially stabilizing score functions satisfying a moment condition. By incorporating stabilization methods into the Malliavin-Stein theory, we obtain rates of normal approximation for sums of stabilizing score functions which either improve upon existing rates or are the first of their kind.Our general rates hold for functionals of marked input on spaces more general than full-dimensional subsets of R d , including m-dimensional Riemannian manifolds, m ≤ d. We use the general results to deduce improved and new rates of normal convergence for several functionals in stochastic geometry, including those whose variances re-scale as the volume or the surface area of an underlying set. In particular, we improve upon rates of normal convergence for the k-face and ith intrinsic volume functionals of the convex hull of Poisson and binomial random samples in a smooth convex body in dimension d ≥ 2. We also provide improved rates of normal convergence for statistics of nearest neighbors graphs and high-dimensional data sets, the number of maximal points in a random sample, estimators of surface area and volume arising in set approximation via Voronoi tessellations, and clique counts in generalized random geometric graphs.

Schreiber and Yukich [Ann. Probab. 36 (2008) establish an asymptotic representation for random convex polytope geometry in the unit ball B d , d ≥ 2, in terms of the general theory of stabilizing functionals of Poisson point processes as well as in terms of generalized paraboloid growth processes. This paper further exploits this connection, introducing also a dual object termed the paraboloid hull process. Via these growth processes we establish local functional limit theorems for the properly scaled radius-vector and support functions of convex polytopes generated by high-density Poisson samples. We show that direct methods lead to explicit asymptotic expressions for the fidis of the properly scaled radius-vector and support functions. Generalized paraboloid growth processes, coupled with general techniques of stabilization theory, yield Brownian sheet limits for the defect volume and mean width functionals. Finally we provide explicit variance asymptotics and central limit theorems for the k-face and intrinsic volume functionals.