Rates of Convergence of Means of Euclidean Functionals

Abstract

Let L be the Euclidean functional with p-th power-weighted edges. Examples include the sum of the p-th power-weighted lengths of the edges in minimal spanning trees, traveling salesman tours, and minimal matchings. Motivated by the works of Steele, Redmond and Yukich (Ann. Appl. Probab. 4, 1057–1073, 1994, Stoch. Process. Appl. 61, 289–304, 1996) have shown that for n i.i.d. sample points {X 1,…,X n } from [0,1]d, L({X 1,…,X n })/n (d−p)/d converges a.s. to a finite constant. Here we bound the rate of convergence of EL({X 1,…,X n })/n (d−p)/d.