Limit theory for the Gilbert graph

For a given homogeneous Poisson point process in Rd two points are connected by an edge if their distance is bounded by a prescribed distance parameter. The behavior of the resulting random graph, the Gilbert graph or random geometric graph, is investigated as the intensity of the Poisson point process is increased and the distance parameter goes to zero. The asymptotic expectation and covariance structure of a class of length-power functionals are computed. Distributional limit theorems are derived that have a Gaussian, a stable or a compound Poisson limiting distribution. Finally, concentration inequalities are provided using the convex distance.