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Random nested tetrahedra

  • Gérard Letac (a1) and Marco Scarsini (a2)
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, …, α n of E. Choose n independent points β1, β2, …, β n 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, …, α n ) (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).

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Corresponding author
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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Advances in Applied Probability
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