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The convex hull of samples from self-similar distributions

Part of: General convexity Geometric probability and stochastic geometry Classical measure theory

Published online by Cambridge University Press:  01 July 2016

Irene Hueter
Irene Hueter*
Affiliation:
University of Florida
*
∗ Postal address: Department of Mathematics, University of Florida, 358 Little Hall, PO Box 118105, Gainesville, FL 32611-8105, USA.
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Abstract

Let X1, X2,… be i.i.d. random points in ℝ2 with distribution ν, and let Nn denote the number of points spanning the convex hull of X1, X2,…,Xn. We obtain lim infn→∞E(Nn)n-1/3 ≥ γ1 and E(Nn) ≤ γ2n1/3(logn)2/3 for some positive constants γ1, γ2 and sufficiently large n under the assumption that ν is a certain self-similar measure on the unit disk. Our main tool consists in a geometric application of the renewal theorem. Exactly the same approach can be adopted to prove the analogous result in ℝd.

Keywords

Convex hullrenewal theoremself-similar

MSC classification

Primary: 28A80: Fractals 60D05: Geometric probability and stochastic geometry
Secondary: 52A07: Convex sets in topological vector spaces
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 31 , Issue 1 , March 1999 , pp. 34 - 47
Copyright
Copyright © Applied Probability Trust 1999 

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References

Bárány, I. and Larman, D. G. (1988). Convex bodies, economic cap coverings, random polytopes. Mathematika 35, 274–291.CrossRefGoogle Scholar
Devroye, L. (1991). On the oscillation of the expected number of extreme points of a random set. Stat. Prob. Lett. 11, 281–286.CrossRefGoogle Scholar
Efron, B. (1965). The convex hull of a random set of points. Biometrika 52, 331–343.CrossRefGoogle Scholar
Falconer, K. (1990). Fractal Geometry. Wiley, Chichester.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn. Wiley, New York.Google Scholar
Hueter, I. (1994). The convex hull of a normal sample. Adv. Appl. Prob. 26, 855–875.CrossRefGoogle Scholar
Hutchinson, J. (1981). Fractals and self-similarity. Indiana Univ. Math. J. 30, 713–747.CrossRefGoogle Scholar
Lalley, S. P. (1990). Travelling salesman with a self-similar itinerary. Prob. Eng. Inform. Sci. 4, 1–18.CrossRefGoogle Scholar
Massé, B., (1993). Principes d'invariance pour la probabilité d'un dilaté de l'enveloppe convexe d'un échantillon. Ann. Inst. Henri Poincaré 29, 37–55.Google Scholar
Rényi, A. and Sulanke, R. (1963). Über die konvexe Hülle von n zufällig gewählten Punkten. Z. Wahrscheinlichkeitsth. 2, 75–84.CrossRefGoogle Scholar
Schief, A. (1994). Separation properties for self-similar sets. Proc. AMS 122, 111–115.CrossRefGoogle Scholar