A STRONG LAW FOR THE LARGEST NEAREST-NEIGHBOUR LINK BETWEEN RANDOM POINTS

By Mathew D. Penrose.

Suppose X₁,X₂,X₃,... 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. Let R_i,k,n denote the distance from X_i to its k-th nearest neighbour amongst the first n points, and let M_n,k = max_{i <= n} R_i,k,n. We derive an almost sure limit for n (M_n,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).