Dirichlet and quasi-Bernoulli laws for perpetuities
Résumé
Let X, B and Y be three Dirichlet, Bernoulli and beta independent random variables such that X ∼ D(a0,. .. , a d), such that Pr(B = (0,. .. , 0, 1, 0,. .. , 0)) = ai/a with a = d i=0 ai and such that Y ∼ β(1, a). We prove that X ∼ X(1 − Y) + BY. This gives the stationary distribution of a simple Markov chain on a tetrahedron. We also extend this result to the case when B follows a quasi Bernoulli distribution B k (a0,. .. , a d) on the tetrahedron and when Y ∼ β(k, a). We extend it even more generally to the case where X is a Dirichlet process and B is a quasi Bernoulli random probability. Finally the case where the integer k is replaced by a positive number c is considered when a0 =. .. = a d = 1.
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