Dirichlet and Quasi-Bernoulli Laws for Perpetuities
Paweł Hitczenko, Gérard Letac · Journal of Applied Probability · 2014
LetX,B, andYbe the Dirichlet, Bernoulli, and beta-independent random variables such thatX~D(a0, …,ad), Pr(B= (0, …, 0, 1, 0, …, 0)) =ai/awitha= ∑i=0dai, andY~ 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 distributionBk(a0, …,ad) withkan integer such that the above result holds whenBfollowsBk(a0, …,ad) and whenY~ β(k,a). We extend it even more generally to the case whereXandBare random probabilities such thatXis Dirichlet andBis quasi-Bernoulli. Finally, the case where the integerkis replaced by a positive numbercis considered whena0= · · · =ad= 1.