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The Smallest Parallelepiped of n Random Points and Peeling

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  • Ju¨rg Hu¨sler

    (University of Bern)

Abstract

Let n random points be given with uniform distribution in the d-dimensional unit cube [0,1]d. The smallest parallelepiped A which includes all the n random points is dealt with. We investigate the asymptotic behavior of the volume of A as n tends to ∞. Using a point process approach, we derive also the asymptotic behavior of the volumes of the k-th smallest parallelepipeds A n (k) which are defined by iteration. Let A n = A n (1) . Given A n (k,-,1) delete the random points X i which are on the boundary ∂A n (k,-,1) , and construct the smallest parallelepiped which includes the inner points of A n (k,-,1) , this defines A n (k) . This procedure is known as peeling of the parallelepiped An.

Suggested Citation

  • Ju¨rg Hu¨sler, 2000. "The Smallest Parallelepiped of n Random Points and Peeling," Methodology and Computing in Applied Probability, Springer, vol. 2(2), pages 169-181, August.
    • Handle: RePEc:spr:metcap:v:2:y:2000:i:2:d:10.1023_a:1010098023091
    DOI: 10.1023/A:1010098023091
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    References listed on IDEAS

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    1. González-Barrios, JoséM., 1996. "The volume of the smallest parallelepiped including m random points," Statistics & Probability Letters, Elsevier, vol. 30(2), pages 139-145, October.
    2. Hashorva, E. & Hüsler, J., 1999. "Extreme Values in FGM Random Sequences," Journal of Multivariate Analysis, Elsevier, vol. 68(2), pages 212-225, February.
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