Asymptotic theory of extreme dual generalized order statistics

In a wide subclass of dual generalized order statistics (dgos) (which contains the most important models of descendingly ordered random variables), when the parameters γ1,…,γn are assumed to be pairwise different, we study the weak convergence of the lower extremes, under general strongly monotone continuous transformations. It is revealed that the weak convergence of the maximum order statistics guarantees the weak convergence of any lower extreme dgos. Moreover, under linear and power normalization and by a suitable choice of these normalizations, the possible weak limits of any rth upper extreme order statistic are the same as the possible weak limits of the rth lower extreme dgos.