A STRONG LAW FOR THE LARGEST NEAREST-NEIGHBOUR LINK BETWEEN RANDOM POINTS
By Mathew D. Penrose.
are independent random points in d-dimensional
space with common density f, having compact
support A with smooth boundary.
Suppose the restriction of
f to A is continuous.
denote the distance from Xi to its k-th nearest
neighbour amongst the first n points, and let
Mn,k = maxi <= n
derive an almost sure limit for n
(Mn,k)d/ log n .
We give an analogous result for the case where the points lie in a compact
smooth d-dimensional Riemannian manifold.
Journal of the London Mathematical Society (2),
60, 951-960 (1999).