Maxima of Dirichlet and triangular arrays of gamma variables

TL;DR

This paper investigates the asymptotic behavior of maxima in triangular arrays of gamma variables and Dirichlet vectors, revealing conditions for convergence to Gumbel and non-Gumbel limits under various parameter regimes.

Contribution

It provides new weak limit results for maxima of gamma arrays and Dirichlet vectors, expanding understanding of their extreme value distributions.

Findings

Weak convergence of maxima under different parameter conditions

Identification of Gumbel and non-Gumbel limit distributions

Extension to maxima of Dirichlet vector coordinates

Abstract

Consider a rowwise independent triangular array of gamma random variables with varying parameters. Under several different conditions on the shape parameter, we show that the sequence of row-maximums converges weakly after linear or power transformation. Depending on the parameter combinations, we obtain both Gumbel and non-Gumbel limits. The weak limits for maximum of the coordinates of certain Dirichlet vectors of increasing dimension are also obtained using the gamma representation.