Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-25T14:24:15.908Z Has data issue: false hasContentIssue false
  • English
  • Français

Invariance principle for the deviation between the probability content and the interior point proportion of a random convex hull

Part of: Geometric probability and stochastic geometry Multivariate analysis General convexity

Published online by Cambridge University Press:  14 July 2016

Bruno Massé
Bruno Massé*
Affiliation:
Université du Littoral, Dunkerque
*
∗Postal address: Université du Littoral, 9, Quai de la Citadelle – B.P. 1022, 59375 – Dunkerque Cedex 1, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

The quadratic mean of the deviation between the probability content and the interior point proportion of a random convex hull in is investigated. We obtain, in particular, an explicit and distribution-independent bound.

Keywords

MULTIDIMENSIONAL EXTREME VALUES

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 52A22: Random convex sets and integral geometry 62H99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 4 , December 1995 , pp. 1041 - 1047
Copyright
Copyright © Applied Probability Trust 1995 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Aldous, D. J., Fristedt, B., Griffin, P. S., Pruitt, W. E. (1991) The number of extreme points in the convex hull of a random sample. J. Appl. Prob. 28, 287–304.CrossRefGoogle Scholar
[2] Brozius, H. and De Haan, L. (1987) On limiting laws for the convex hull of a sample. J. Appl. Prob. 24, 852–862.CrossRefGoogle Scholar
[3] Buchta, C. (1986) A note on the volume of a random polytope in a tetrahedron. Ill. J. Math. 30, 653–659.Google Scholar
[4] Devroye, L. P. (1991) On the oscillation of the expected number of extreme points of a random set. Statist. Prob. Lett. 11, 281–286.CrossRefGoogle Scholar
[5] Dwyer, R. A. (1988) On the convex hull of random points in a polytope. J. Appl. Prob. 25, 688–699.CrossRefGoogle Scholar
[6] Efron, B. (1965) The convex hull of a random set of points. Biometrika 52, 331–343.CrossRefGoogle Scholar
[7] Geffroy, J. (1964) Sur un problème d'estimation géométrique. Pub. ISUP 13, 191–200.Google Scholar
[8] Groeneboom, P. (1988) Limit theorems for convex hulls. Prob. Theory. Rel. Fields 79, 327–368.CrossRefGoogle Scholar
[9] Grünbaum, B. (1967) Convex Polytopes. Wiley, London.Google Scholar
[10] Massé, B. (1993) Principes d'invariance pour la probabilité d'un dilaté de l'enveloppe convexe d'un échantillon. Ann. Inst. H. Poincaré Prob.-Stat. 29, 37–55.Google Scholar
[11] Reed, W. J. (1974) Random points in a simplex. Pacific J. Math. 54, 183–198.CrossRefGoogle Scholar
[12] Rényi, A. and Sulanke, R. (1964) über die konvexe Hülle von n zuffällig gewählten Punkten. Z. Wahrscheinlichkeitsth. 3, 138–147.CrossRefGoogle Scholar
[13] Schneider, R. (1988) Random approximation of convex sets. J. Microscopy 151, 211–227.CrossRefGoogle Scholar
[14] Wel, B. F. Van (1989) The convex hull of a uniform sample from the interior of a simple d-polytope. J. Appl. Prob. 26, 259–273.Google Scholar