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Rates of Convergence of Means of Euclidean Functionals

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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.

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Authors and Affiliations

  1. Department of Mathematics, Sungkyunkwan University, Suwon, Korea, 440-746

    Yooyoung Koo

  2. Department of Mathematics, Yonsei University, Seoul, Korea, 120-749

    Sungchul Lee

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  1. Yooyoung Koo
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  2. Sungchul Lee
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Correspondence to Sungchul Lee.

Additional information

Y. Koo supported by the BK21 project of the Department of Mathematics, Sungkyunkwan University.

S. Lee supported by the BK21 project of the Department of Mathematics, Yonsei University.

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Koo, Y., Lee, S. Rates of Convergence of Means of Euclidean Functionals. J Theor Probab 20, 821–841 (2007). https://doi.org/10.1007/s10959-007-0089-7

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  • DOI: https://doi.org/10.1007/s10959-007-0089-7

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