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, Yukich (1994, 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 .
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