| Let cPon := {lcub}X1,..., XPon {rcub} be i.i.d. random points in Rd where Pon is an independent Poisson random variable with mean n. Recently Penrose [18] and Baryshnikov and Yukich [4] proved that under suitable conditions the finite dimensional distributions of re-normalized random point measures converge to a Gaussian field. These random point measures are defined in terms of a functional xi which acts on the random point set cPon . When the Xi have valued in [0,1] d I extend these results to show convergence of re-normalized centered random point measures as a process in D ([0,1] d). Additionally I consider the directed and undirected nearest neighbors graph on a collection of Pon points which are uniformly distributed on the Cantor set. I prove convergence to a constant of the re-scaled expected total edge length of this random graph. The re-scaling factor is a function of the fractal dimension and has a log periodic, non-constant behavior. |
