Advertisement

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)

Abstract

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.

Keywords

Convex Hull Lebesgue Measure Extreme Point Random Vector Empirical Probability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Gaenssler, P. and Stute, W. (1975/76). A survey on some results for empirical processes in the i.i.d. case. RUB-Preprint Series No. 15.Google Scholar
  2. [2]
    Gaenssler, P. and Stute, W. (1976). On uniform convergence of measures with applications to uniform convergence of empirical distributions. In this volume.Google Scholar
  3. [3]
    Rao, R.R. (1962). Relations between weak and uniform convergence of measures with appl. Ann. Math. Statist. 33, 659–680.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Stute, W. (1976). On a generalization of the Glivenko-Cantelli theorem. Z. Wahrscheinlichkeitstheorie verw. Gebiete 35, 167–175MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Zaremba, S.K. (1970). La discrépance isotrope et l'intégration numérique. Ann. Mat. Pura Appl. (IV) 87, 125–135.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1976

Authors and Affiliations

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

Personalised recommendations