Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Limit theorems for functionals of convex hulls
Download PDF
Download PDF
  • Published: March 1994

Limit theorems for functionals of convex hulls

  • A. J. Cabo1 &
  • P. Groeneboom1 

Probability Theory and Related Fields volume 100, pages 31–55 (1994)Cite this article

  • 133 Accesses

  • 38 Citations

  • Metrics details

Summary

In [4] a central limit theorem for the number of vertices of the convex hull of a uniform sample from the interior of a convex polygon is derived. This is done by approximating the process of vertices of the convex hull by the process of extreme points of a Poisson point process and by considering the latter process of extreme points as a Markov process (for a particular parametrization). We show that this method can also be applied to derive limit theorems for the boundary length and for the area of the convex hull. This extents results of Rényi and Sulanke (1963) and Buchta (1984), and shows that the boundary length and the area have a strikingly different probabilistic behavior.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Buchta, C.: Stochastische Approximation konvexer Polygone. Z. Wahrscheinlichkeitstheor. Verw. Geb.67, 283–304 (1984)

    Google Scholar 

  2. Efron, B.: The convex hull of a random set of points Biometrika52, 331–343 (1965)

    Google Scholar 

  3. Feller, W.: Introduction to Probability Theory and its Applications (2nd ed.) New York: Wiley 1971

    Google Scholar 

  4. Groeneboom, P.: Limit Theorems for Convex Hulls. Probab. Theory Relat. Fields79, 327–368 (1988)

    Google Scholar 

  5. Hueter, I.: The convex hull ofn random points and its vertex process. Doctoral Dissertation. University of Berne, 1992

  6. Ibragimov, I.A., Linnik, Y.U.: Independent and stationary sequences of random variables. Groningen: Wolters Noordhoff 1971

    Google Scholar 

  7. Rényi, A., Sulanke, R.: Über die konvexe Hülle vonn zufällig gewählten Punkten. Z. Wahrscheinlichkeitstheor. Verw. Geb.2, 75–84 (1963)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, Delft University of Technology, Mekelweg 4, 2628 CD, Delft, The Netherlands

    A. J. Cabo & P. Groeneboom

Authors
  1. A. J. Cabo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. Groeneboom
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Cabo, A.J., Groeneboom, P. Limit theorems for functionals of convex hulls. Probab. Th. Rel. Fields 100, 31–55 (1994). https://doi.org/10.1007/BF01204952

Download citation

  • Received: 25 June 1991

  • Revised: 04 February 1994

  • Issue Date: March 1994

  • DOI: https://doi.org/10.1007/BF01204952

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Mathematics Subject Classifications (1991)

  • 52A22
  • 52A38
  • 60G44
  • 60G55
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature