Successive maxima of samples from a GEM distribution

TL;DR

This paper analyzes the behavior of maximum values in samples from GEM distributions, revealing their growth rates and limiting distributions as sample size increases.

Contribution

It provides new results on the distribution and growth rate of maxima in GEM$( heta)$ and GEM$(eta, heta)$ distributions, including explicit limiting distributions.

Findings

Maxima in GEM$( heta)$ grow as $ heta ext{log}(n)$ with sample size.

Maxima in GEM$(eta, heta)$ grow as a random factor of $n^{eta/(1-eta)}$.

Limiting distributions for the maxima are explicitly derived.

Abstract

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