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WEAK LAWS IN GEOMETRIC PROBABILITY

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By Mathew D. Penrose and J.E. Yukich

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.