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Probability and Approximation
May 18, 2022 18:00, Smolensk, Smolensk, SmolSU, Distant form
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Maximal values in a sample from a GEM distribution and interleaving point processes
Yu. V. Yakubovich Saint Petersburg State University
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Abstract:
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.
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