A necessary condition for the convergence of the isotrope discrepancy

  • W. Stute
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 566)


Given a sequence (ξi)i∈N of independent identically distributed (i.i.d.) ℝk-valued random vectors with distribution \(\mu = Q_{\xi _1 }\), the isotrope discrepancy D n μ (ω) is defined by
, where μ n ω denotes the empirical probability distribution and the supremum is taken over the class ℓk of all convex measurable subsets of ℝk. In the present paper it is proved that μc(e(C))=0 for all C ε ℓk whenever D n μ (ω)→0 a.s., and where e(C) denotes the set of all extreme points of C.


Convex Hull Lebesgue Measure Extreme Point Random Vector Empirical Probability 
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Copyright information

© Springer-Verlag 1976

Authors and Affiliations

  • W. Stute
    • 1
  1. 1.Math. Inst. NARuhr UniversityBochumWest Germany

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