2012/04/11 by Paweł Hitczenko, Pawel Hitczenko, Hitczenko, Pawel +3
Computer Science · Decision Sciences · Mathematics · #60E99 #60J05 #Bayesian Methods and Mixture Models #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Statistical Distribution Estimation and Applications #math.PR #msc:60E99 #msc:60J05
paper · pdf · doi:10.48550/arxiv.1204.2315
18 pages
arxiv created 2012/04/11 · openalex publication_date 2012/04/11 · arxiv updated 2012/04/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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=∑i=0dai and such that Y∼ β(1,a). We prove that X∼ X(1-Y)+BY. This gives the stationary distribution of a simple Markov chain on a tetrahedron. We also extend this result to the case when B follows a quasi Bernoulli distribution Bk(a0,...,ad) on the tetrahedron and when Y∼ β(k,a). We extend it even more generally to the case where X is a Dirichlet process and B is a quasi Bernoulli random probability. Finally the case where the integer k is replaced by a positive number c is considered when a0=...=ad=1. Keywords Perpetuities, Dirichlet process, Ewens distribution, quasi Bernoulli laws, probabilities on a tetrahedron, Tc transform, stationary distribution. AMS classification 60J05, 60E99.