On the fraction of random points by specified nearest-neighbour interrelations and degree of attraction

Let Z1, ., Zn be i.i.d. random vectors (.points') defined in having common density f(x) that is assumed to be continuous almost everywhere. For a fixed but otherwise arbitrary norm |.| on , consider the fraction Vn of those points Z1, ., Zn that are the lth nearest neighbour (with respect to |.|) to their own kth nearest neighbour, and write Sn for the fraction of points that are the nearest neighbour of exactly k other points. We derive the stochastic limits of Vn and Sn, as n tends to., and show how the results may be applied to the multivariate non-parametric two-sample problem.