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, . . . , a_d), Pr(B = (0, . . . , 0, 1, 0, . . . , 0)) = a_i / a with a = ∑_i=0^da_i, 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 B_k(a₀, . . . , a_d) with k an integer such that the above result holds when B follows B_k(a₀, . . . , a_d) 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 = · · · = a_d = 1.