Successive maxima of samples from a GEM distribution

We show that the maximal value in a size $n$ sample from GEM$(θ)$ distribution is distributed as a sum of independent geometric random variables. This implies that the maximal value grows as $θ\log(n)$ as $n\to\infty$. For the two-parametric GEM$(α,θ)$ distribution we show that the maximal value grows as a random factor of $n^{α/(1-α)}$ and find the limiting distribution.

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