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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.
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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 

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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