Perpetuity property of the Dirichlet distribution

TL;DR

This paper demonstrates a perpetuity property of the Dirichlet distribution, revealing its stationary distribution in a Markov chain on a tetrahedron and extending to Dirichlet processes and quasi Bernoulli distributions.

Contribution

It introduces a novel perpetuity property of the Dirichlet distribution and extends it to Dirichlet processes and quasi Bernoulli distributions, with applications to Markov chains.

Findings

Proves the Dirichlet distribution's perpetuity property.

Identifies the stationary distribution of a Markov chain on a tetrahedron.

Extends results to Dirichlet processes and quasi Bernoulli distributions.

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 $\Pr(B=(0,...,0,1,0,...,0))=a_i/a$ with $a=\sum_{i=0}^da_i$ 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.