Skip to main content

A necessary condition for the convergence of the isotrope discrepancy

  • 342 Accesses

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 μn (ω) is defined by

, where μ ωn 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 μn (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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   29.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   39.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  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. 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. Rao, R.R. (1962). Relations between weak and uniform convergence of measures with appl. Ann. Math. Statist. 33, 659–680.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. Stute, W. (1976). On a generalization of the Glivenko-Cantelli theorem. Z. Wahrscheinlichkeitstheorie verw. Gebiete 35, 167–175

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. Zaremba, S.K. (1970). La discrépance isotrope et l'intégration numérique. Ann. Mat. Pura Appl. (IV) 87, 125–135.

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1976 Springer-Verlag

About this paper

Cite this paper

Stute, W. (1976). A necessary condition for the convergence of the isotrope discrepancy. In: Gaenssler, P., Révész, P. (eds) Empirical Distributions and Processes. Lecture Notes in Mathematics, vol 566. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0096884

Download citation

  • DOI: https://doi.org/10.1007/BFb0096884

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-08061-9

  • Online ISBN: 978-3-540-37515-9

  • eBook Packages: Springer Book Archive