Factorial moment densities of Poisson-Dirichlet components (${\alpha }_{\theta }^{(m)}(dx)={\theta }^m\frac{(1-{\sum }_i{x}_i{)}^{\theta -1}}{{\prod }_i{x}_i}dx$)

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.