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The convex hull of samples from self-similar distributions

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

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 X 1, X 2,… be i.i.d. random points in ℝ2 with distribution ν, and let N n denote the number of points spanning the convex hull of X 1, X 2,…,X n . We obtain lim infn→∞E (N n )n -1/3 ≥ γ1 and E (N n ) ≤ γ2 n 1/3(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

Information

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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