Perpetuity property of the Dirichlet distribution

Abstract

Let $X$, $B$ and $Y$ be three Dirichlet, Bernoulli and beta independent random variables such that $X\sim \mathcal{D}(a_0,...,a_d),$ such that $\mathrm{Pr}(B=(0,...,0,1,0,...,0))=a_i/a$ with $a={\sum }_{i=0}^d{a}_i$ and such that $Y\sim \beta (1,a).$ We prove that $X\sim X(1-Y)+BY.$ 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 ${\mathcal{B}}_k(a_0,...,a_d)$ on the tetrahedron and when $Y\sim \beta (k,a)$. 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 $a_0=...=a_d=1.$ \textsc{Keywords} \textit{Perpetuities, Dirichlet process, Ewens distribution, quasi Bernoulli laws, probabilities on a tetrahedron, $T_c$ transform, stationary distribution.} AMS classification 60J05, 60E99.

18 pages

References in corpus (1)

• On the posterior distribution of classes of random means