Poisson-Dirichlet distribution

In probability theory, Poisson-Dirichlet distributions are probability distributions on the set of nonnegative, non-increasing sequences with sum 1, depending on two parameters ${\displaystyle \alpha \in [0,1)}$ and ${\displaystyle \theta \in (-\alpha ,\infty )}$. It can be defined as follows. One considers independent random variables ${\displaystyle (Y_{n})_{n\geq 1}}$ such that ${\displaystyle Y_{n}}$ follows the beta distribution of parameters ${\displaystyle 1-\alpha }$ and ${\displaystyle \theta +n\alpha }$. Then, the Poisson-Dirichlet distribution ${\displaystyle PD(\alpha ,\theta )}$ of parameters ${\displaystyle \alpha }$ and ${\displaystyle \theta }$ is the law of the random decreasing sequence containing ${\displaystyle Y_{1}}$ and the products ${\displaystyle Y_{n}\prod _{k=1}^{n-1}(1-Y_{k})}$. This definition is due to Jim Pitman and Marc Yor.^[1][2] It generalizes Kingman's law, which corresponds to the particular case ${\displaystyle \alpha =0}$.^[3]