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Random nested tetrahedra

Part of: General convexity Limit theorems Geometric probability and stochastic geometry Distribution theory - Probability

Published online by Cambridge University Press:  01 July 2016

Gérard Letac  and
Marco Scarsini
Gérard Letac*
Affiliation:
Université Paul Sabatier
Marco Scarsini*
Affiliation:
Univerità D'Annunzio
*
∗ Postal address: Laboratoire de Statistique et Probabilités, Université Paul Sabatier, 31062 Toulouse, France. Email address: letac@cict.fr
∗∗ Postal address: Dipartimento di Scienze, Univerità D'Annunzio, 65127 Pescara, Italy. Email address: scarsini@sci.unich.it
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Abstract

In a real n-1 dimensional affine space E, consider a tetrahedron T0, i.e. the convex hull of n points α1, α2, …, αn of E. Choose n independent points β1, β2, …, βn randomly and uniformly in T0, thus obtaining a new tetrahedron T1 contained in T0. Repeat the operation with T1 instead of T0, obtaining T2, and so on. The sequence of the Tk shrinks to a point Y of T0 and this note computes the distribution of the barycentric coordinates of Y with respect to (α1, α2, …, αn) (Corollary 2.3). We also obtain the explicit distribution of Y in more general cases. The technique used is to reduce the problem to the study of a random walk on the semigroup of stochastic (n,n) matrices, and this note is a geometrical application of a former result of Chamayou and Letac (1994).

Keywords

Random subdivisionlimit distributionsDirichlet distributioncompositional databarycentric coordinatesproducts of random matrices

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 52A22: Random convex sets and integral geometry
Secondary: 60E05: Distributions 60F99: None of the above, but in this section
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 30 , Issue 3 , September 1998 , pp. 619 - 627
Copyright
Copyright © Applied Probability Trust 1998 

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