The central limit theorem for Euclidean minimal spanning trees I

Abstract

Let {Xi: i ≥ 1} be i.i.d. with uniform distribution [- 1/2, 1/2]d, d ≥ 2, and let Tn be a minimal spanning tree on {X1,..., Xn}. For each strictly positive integer α, let N({X1,..., Xn}; α) be the number of vertices of degree α in Tn. Then, for each α such that P(N({X1,..., Xα+1}; α) = 1) > 0, we prove a central limit theorem for N({X1,..., Xn}; α).