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Quasi-random points keep their distance

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  • Sobol, I.M.
  • Shukhman, B.V.

Abstract

In contrast to random points that may cluster, quasi-random points keep their distance. These distances are investigated.1.If N independent random points in the n-dimensional unit hypercube are selected, two of these points may be arbitrarily close. However, if Q0, Q1, …, QN−1, are quasi-random points, the minimum distance between pairs of these points, dN, has a positive lower bound. For the Sobol sequence dN≥1/2nN−1. Numerical experiments suggest that for large N(1)dN≍N−1/n.2.For certain search algorithms, it is important to know points Qi and Qi+1 that are not close. For the Sobol sequence, the distancesρ(Q2k,Q2k+1)=12n,and14n≤ρ(Q4k+1,Q4k+2)≤145n+c,where c=0 for even n and c=4 for odd n.3.Numerical estimations of dN for the Halton and Faure sequences were carried out. It is likely that for these sequences, (1) is true also.

Suggested Citation

  • Sobol, I.M. & Shukhman, B.V., 2007. "Quasi-random points keep their distance," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 75(3), pages 80-86.
    • Handle: RePEc:eee:matcom:v:75:y:2007:i:3:p:80-86
    DOI: 10.1016/j.matcom.2006.09.004
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    References listed on IDEAS

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    1. Spassimir H. Paskov & Joseph F. Traub, 1995. "Faster Valuation of Financial Derivatives," Working Papers 95-03-034, Santa Fe Institute.
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