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Preprint
Perpetuity property of the Dirichlet distribution
10 Apr 2012
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.
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Details
- Title
- Perpetuity property of the Dirichlet distribution
- Creators
- Pawel HitczenkoGerard Letac
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991020531970304721