Rate of convergence for the Euclidean minimum spanning tree limit law

The minimum spanning tree problem (MSTP) in the plane requires finding the length of the shortest tree spanning n points of R 2. We are concerned here with stochastic versions of the problem. First let Xi, 1 _ i < o0, be uniformly and independently distributed random variables in [0,1] 2 and let LMsT(n) be the length of the shortest tree spanning {X 1, X 2 . . . . . Xn}. Steele [9] proved that LMsT(n) is asymptotic to /3MSTX/n with probability one (the same being true in expectation). In fact this result is valid for any uniform i.i.d, random variables with compact support of measure one in R d, d ~ 2, provided v~ is replaced by n (dt)/d, the constant depending only on the dimension of the space and not on the shape of the compact support. Similar results had previously been obtained for the traveling salesman problem, the weighted matching problem, and the Steiner tree problem (Beardwood et al.