Weak laws of large numbers in geometric probability
The authors use a coupling argument to establish a general weak law of large numbers for functionals of binomial point processes in \(d\)-dimensional Euclidean space. The limit depends explicitly on the density, possibly non-uniform, of the process. When the point set forms the vertex set of a graph, functionals of interest include total edge length with arbitrary weighting, number of edges and number of components. The general result is applied to various graphs. Weak laws of large numbers are also obtained for functionals of marked point processes, including statistics of Boolean models. The methods were used previously to obtain a strong law of large numbers for functionals on uniform point sets; see e.g. the authors [Ann. Appl. Probab. 12, 272-301 (2002; Zbl 1018.60023)].
scientific article; zbMATH DE number 1906136