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Random nested tetrahedra

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

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 T 0, i.e. the convex hull of n points α1, α2, …, αn of E. Choose n independent points β1, β2, …, βn randomly and uniformly in T 0, thus obtaining a new tetrahedron T 1 contained in T 0. Repeat the operation with T 1 instead of T 0, obtaining T 2, and so on. The sequence of the T k shrinks to a point Y of T 0 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

Information

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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