Asymptotics for Euclidean functionals with power-weighted edges

We provide general and relatively simple conditions under which Euclidean functionals Lp on [0, 1]d with pth power-weighted edges satisfy the limit limn→∞ Lp(X1,…,Xn)/nd−pd = β ∫[0,1]dƒ(x)d−pddx c.c., Xi, i ≥ 1, are i.i.d. random variables with values in [0, 1]d, 0 < p < d, β:=β(Lp, d) is a constant, f is the density of the absolutely continuous part of the law of X1, and c.c denotes complete convergence. This general result is shown to apply to the minimal spanning tree, shortest tour, and minimal matching functionals. The approach provides a rate of convergence for the power-weighted minimal spanning tree functional, resolving a question raised by Steele (1988).