Dirichlet and Quasi-Bernoulli Laws for Perpetuities

Abstract

Let X, B, and Y be the Dirichlet, Bernoulli, and beta-independent random variables such that X ~ D(a0, …, ad), Pr(B = (0, …, 0, 1, 0, …, 0)) = ai / a with a = ∑i=0dai, and Y ~ β(1, a). Then, as proved by Sethuraman (1994), X ~ X(1 - Y) + BY. This gives the stationary distribution of a simple Markov chain on a tetrahedron. In this paper we introduce a new distribution on the tetrahedron called a quasi-Bernoulli distribution Bk(a0, …, ad) with k an integer 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.