A Strong Law for the Largest Nearest-Neighbour Link between Random Points
Mathew D. Penrose · Journal of the London Mathematical Society · 1999
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∣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⩽n Rni, k. Let θ denote the volume of the unit ball. Then as n → ∞, n θ M n , k d / log n → max ( ( min f Ω ) - 1 , 2 ( 1 - 1 / d ) ( min f ∂ Ω ) - 1 ) , almost surely . If instead the points lie in a compact smooth d-dimensional Riemannian manifold K, then nθMdn, k/log n → (minKf)−1, almost surely.