Maxima of Dirichlet and triangular arrays of gamma variables

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.