The central limit theorem for Euclidean minimal spanning trees. I (Q1379719)

Let \(X_i\), \(i= 1,2,\dots, n\), be i.i.d. r.v.s with uniform distribution on \(d\)-dimensional square of \([-1/2,1/2]\) and let \(T_n\) be a minimal spanning tree on \(\{X_1,\dots, X_n\}\). For each strictly positive integer \(m\), let \(N(\{X_1,\dots, X_n\};m)\) be the number of vertices of degree \(m\) in \(T_n\). Then, for each \(m\) such that \(P(N(\{X_1,\dots, X_{m+ 1}\};m)= 1)>0\), the author proves a central limit theorem for \(N(\{X_1,\dots, X_n\};m)\). He also proves similar CLT for Poisson point process of density \(n\) on \([-1/2,1/2]^d\), \(d\geq 2\).