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Rates of Convergence of Means of Euclidean Functionals

Author

Listed:
  • Yooyoung Koo

    (Sungkyunkwan University)

  • Sungchul Lee

    (Yonsei University)

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 .

Suggested Citation

  • Yooyoung Koo & Sungchul Lee, 2007. "Rates of Convergence of Means of Euclidean Functionals," Journal of Theoretical Probability, Springer, vol. 20(4), pages 821-841, December.
    • Handle: RePEc:spr:jotpro:v:20:y:2007:i:4:d:10.1007_s10959-007-0089-7
    DOI: 10.1007/s10959-007-0089-7
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    References listed on IDEAS

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    1. Redmond, C. & Yukich, J. E., 1996. "Asymptotics for Euclidean functionals with power-weighted edges," Stochastic Processes and their Applications, Elsevier, vol. 61(2), pages 289-304, February.
    2. Lee, Sungchul, 1999. "Asymptotics of power-weighted Euclidean functionals," Stochastic Processes and their Applications, Elsevier, vol. 79(1), pages 109-116, January.
    3. Lee, Sungchul, 2000. "Rate of convergence of power-weighted Euclidean minimal spanning trees," Stochastic Processes and their Applications, Elsevier, vol. 86(1), pages 163-176, March.
    4. J. Michael Steele, 1981. "Complete Convergence of Short Paths and Karp's Algorithm for the TSP," Mathematics of Operations Research, INFORMS, vol. 6(3), pages 374-378, August.
    5. J. Michael Steele, 1990. "Probabilistic and Worst Case Analyses of Classical Problems of Combinatorial Optimization in Euclidean Space," Mathematics of Operations Research, INFORMS, vol. 15(4), pages 749-770, November.
    Full references (including those not matched with items on IDEAS)

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