Dirichlet and Quasi-Bernoulli Laws for Perpetuities

Paweł Hitczenko, Gérard Letac
2014 Journal of Applied Probability  
Let X, B, and Y be the Dirichlet, Bernoulli, and beta-independent random variables such that X ~ D (a 0, ..., a d ), Pr(B = (0, ..., 0, 1, 0, ..., 0)) = a i / a with a = ∑ i=0 d a 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 (a 0, ..., a d ) with k an integer such that the
more » ... r such that the above result holds when B follows B k (a 0, ..., 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 a 0 = · · · = a d = 1.
doi:10.1239/jap/1402578633 fatcat:mvk2dsbhsra4to4ua5zzczlode