Maximal values in a sample from a GEM distribution and interleaving point processes

The talk is devoted to a distribution of a maximal value in a finite sample from a GEM distribution. The GEM distribution is a random partition of the unit interval; its distribution is parameterized by two parameters $\alpha$ and $\theta$. This random partition and its properties will be described in the talk. One can also interpret it as a random discrete distribution on natural numbers. A sample from it is exchangeable. We also consider a sample from a more general random discrete distribution which is obtained by a “stick-breaking” construction. It will also be explained in the talk.
In a simpler case of sampling from the GEM distribution with $\alpha=0$ we are able to describe a distribution of a maximum of n such random variables as a sum of n independent geometric random variables. In a more tricky case $\alpha>0$ such representation, perhaps, does not exist. Yet we can show that the maximum of n samples behaves asymptotically as $n^{\alpha/(1-\alpha)}$ up to a random factor whose distribution is explicitly described.
The talk is based on the joint works of the author with Jim Pitman.