Let Z
1, …, 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 Z
1, …, 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.