Dirichlet and Quasi-Bernoulli Laws for Perpetuities

Let X, B and Y be three Dirichlet, Bernoulli and beta independent random variables such that X ∼ D(a0, . . . , ad), such that Pr(B = (0, . . . , 0, 1, 0, . . . , 0)) = ai/a with a = Pd i=0 ai and such that Y ∼ β(1, a). Then, as was proved by J. Sethuraman [11], X ∼ X(1 − Y ) + BY. This gives the stationary distribution of a simple Markov chain on a tetrahedron. If k is an integer, the present paper introduces a new distribution on the tetrahedron called a quasi Bernoulli distribution Bk(a0, . . . , ad) such that the above result holds when B follows Bk(a0, . . . , ad) and when Y ∼ β(k, a). We extend it even more generally to the case where X and B are random probabilities such that X is Dirichlet and B is quasi Bernoulli. Finally the case where the integer k is replaced by a positive number c is considered when a0 = . . . = ad = 1.