A Strong Law for the Largest Nearest-Neighbour Link between Random Points

Abstract

Suppose that X1, X2, X3, … are independent random points in Rd with common density f, having compact support Ω with smooth boundary ∂Ω, with f[mid ]Ω continuous. Let Rni, k denote the distance from Xi to its kth nearest neighbour amongst the first n points, and let Mn, k = maxi[les ]nRni, k. Let θ denote the volume of the unit ball. Then as n → ∞,formula hereIf instead the points lie in a compact smooth d-dimensional Riemannian manifold K, then nθMdn, k/ log n → (minKf)−1, almost surely.