Growth Rates of Euclidean Minimal Spanning Trees With Power Weighted Edges

Let Xi, 1 ≤ i < ∞, denote independent random variables with values in Rd, d ≥ 2, and let Mn denote the cost of a minimal spanning tree of a complete graph with vertex set {X1, X2, . . . , Xn}, where the cost of an edge (Xi, Xj) is given by ⋺(|Xi - Xj|). Here |Xi - Xj| denotes the Euclidean distance between Xi and Xj and ⋺ is a monotone function. For bounded random variables and 0 < a < d, it is proved that as n → ∞ one has Mn ~ c(a, d)n(d-a)/d∫Rdf(x)(d-a)/d dx with probability 1, provided ⋺(x) ~ xa as x → 0. Here f(x) is the density of the absolutely continuous part of the distribution of the {Xi}.