A STRONG LAW FOR THE LONGEST EDGE OF THE MINIMAL SPANNING TREE

By Mathew D. Penrose.

Suppose X₁,X₂,X₃,... are independent random points in d-dimensional space with common density f, having compact connected support A with smooth boundary. Suppose the restriction of f to A is continuous. Let M_n be the smallest r such that the union of balls of diameter r centred at the first n points is connected. We derive an almost sure limit for n (M_n)^d/ log n .

Annals of Probability 27, 290-298 (1999).