On the value of a random minimum spanning tree problem

Abstract Suppose we are given a complete graph on n vertices in which the lenghts of the edges are independent identically distributed non-negative random variables. Suppose that their common distribution function F is differentiable at zero and D = F′ (0) > 0 and each edge length has a finite mean and variance. Let Ln be the random variable whose value is the length of the minimum spanning tree in such a graph. Then we will prove the following: limn → ∞E(Ln) = ζ(3)/D where ζ(3) = Σk = 1∞ 1/k3 = 1.202… and for any e > 0 limn → ∞ Pr(|Ln− ζ(3)/D|) > e) = 0.