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 ₁, X₂, …, Xₙ\, where the cost of an edge (Xᵢ, Xⱼ) is given by ψ(|Xᵢ - Xⱼ|). Here |Xᵢ - Xⱼ| denotes the Euclidean distance between Xᵢ and Xⱼ and ψ is a monotone function. For bounded random variables and 0 < α < d, it is proved that as n→∞ one has Mₙ ~ c(α, d)n(d - α)/d ∫Rᵈ f(x)(d-α)/d dx with probability 1, provided ψ(x) ~ x^α as x→ 0. Here $f(x)$ is the density of the absolutely continuous part of the distribution of the ᵢ\.