  Asymptotic properties of a vector of length power functionals of random
geometric graphs are investigated. More precisely, its asymptotic covariance
matrix is studied as the intensity of the underlying homogeneous Poisson point
process increases. This includes a systematic discussion of matrix properties
like rank, definiteness, determinant, eigenspaces or decompositions of
interest. For the formulation of the results a case distinction is necessary.
Indeed, in the three possible regimes the respective covariance matrix is of
quite different nature which leads to different statements. Finally, stochastic
consequences for random geometric graphs are derived.
