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).