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 < < d, it is proved that as n one has Mₙ c (, d) n^ (d -) /d ₑ㵧 f (x) ^ (d-) /d dx with probability 1, provided (x) x^ as x 0. Here f (x) is the density of the absolutely continuous part of the distribution of the \Xᵢ\.