Weak laws of large numbers in geometric probability

Abstract

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.

The Objective Method: Probabilistic Combinatorial Optimization and Local Weak Convergence

“…Finally, we should note that the proof of Theorem 8 given in Penrose and Yukich [53] calls on an interesting coupling argument that is likely to have further applications. The key observation is that one may simulate samples that closely approximate the samples from a general density f with help from a Cox process that has a random but conditionally uniform intensity.…”

“…Nevertheless, we hope to provide a useful up-date of the survey of Steele [60]. In particular, we address progress on the minimal spanning tree problem, including a recent weak law of large numbers developed in Penrose and Yukich [53] which has close ties to the objective method. The section also contrasts the objective method and the subadditive method that has served for years as a principal workhorse in the probability theory of Euclidean combinatorial optimization.…”

“…When ξ is stable for almost every realization of any homogeneous Poisson process, these sums typically satisfy a weak of large numbers; in fact, this notion of stability was introduced in Penrose and Yukich [53] for the purpose of framing such weak laws. The next theorem is typical of the results obtained there.…”

“…For example, even the stability of the MST is not easy without the help of some powerful tools. For example, Penrose and Yukich [53] prove stability MST with help from the idea of a blocking set, which is a set of points that effectively isolates the MST in a neighborhood of the origin from any influence by points outside of a certain large disk. This basic structure was introduce by Kesten and Lee [40] (cf.…”

Limit theory for the random on‐line nearest‐neighbor graph

“…Many aspects of the large-sample asymptotic theory for such graphs, which are locally determined in a certain sense, are by now quite well understood. See for example [10,13,14,16,17,23,26].…”