A Strong Law for the Longest Edge of the Minimal Spanning Tree

Abstract

Suppose $X_1, X_2, X_3,\ldots$ are independent random points in $\mathbf{R}^d,d\geq 2$, with common density $f$, having connected compact support $\Omega$ with smooth boundary $\partial\Omega$, with $f|_{\Omega}$ continuous. Let $M_{n}$ denote the smallest $r$ such that the union of balls of diameter $r$ centered at the first $n$ points is connected. Let $\theta$ denote the volume of the unit ball. Then as $n\to\infty$, $$n\theta M^d_n/\log n \to \max\Big(\big(\min\limits_{\Omega}f\big)^{-1},2(1 - 1/d)\big(\min\limits_{\partial\Omega}f\big)^{-1}\Big),\quad\text{a.s.}$$