One-parameter Poisson-Dirichlet distribution ($PD(\theta )=PD(0,\theta )$)

Take a Poisson random measure on $(0,\infty )$ with intensity $\theta e^{-u}du/u$ and arrange its atoms as $J_1>J_2>\cdots$. The sum $T={\sum }_i{J}_i$ is finite and positive almost surely, has Gamma distribution with shape $\theta$ and rate one, and the law of $(J_i/T)$ is $PD(\theta )$. The order is decreasing, so the first component is the largest; a size-biased selection of a component has a different law.

For $a\ge 1/2$, at most one component exceeds $a$, so $1-F_{\theta }(a)=\theta {\int }_a^1{x}^{-1}(1-x{)}^{\theta -1}dx$. Below one half several components can exceed the threshold. Put $N_a=\#\{i:p_i>a\}$ and use the finite identity $1_{\{N_a=0\}}={\sum }_m(-1{)}^m(N_a{)}_m/m!$, with the factorial moment densities of Poisson-Dirichlet components. It gives the full probability distribution as finite inclusion-exclusion over $x_i>a$, ${\sum }_i{x}_i<1$. In contrast, a size-biased component is beta-distributed.

This density is supported where all $x_i>0$ and ${\sum }_i{x}_i<1$, and gives the expected sum over ordered distinct components of $PD(\theta )$. Apply the iterated Mecke formula for a Poisson random measure to $m$ selected jumps $u_i$ and the remaining gamma total $t$. With $s=t+{\sum }_i{u}_i$ and $x_i=u_i/s$, the Jacobian is $s^m$ and the integral in $s$ is $\Gamma (\theta )$, giving the density. It describes joint counts of large components, beyond the first-moment measure.

For $PD(\theta )$, $\mathbb{E}{\sum }_i\phi (p_i)=\int \phi d{\alpha }_{\theta }$ whenever the absolute integral is finite. Apply the Mecke formula for a Poisson random measure to a selected jump $u$ and the remaining total $t$, whose density is $t^{\theta -1}{e}^{-t}/\Gamma (\theta )$. Changing variables to $s=u+t$ and $x=u/s$ factors out a Gamma function and gives the displayed measure. Weighting it by $x$ gives a probability density $\theta (1-x{)}^{\theta -1}$, the Beta distribution of a size-biased component.