On the convergence of extreme distributions under power normalization
This paper deals with the convergence in distribution of the maximum of $n$ independent and identically distributed random variables under power normalization. We measure the difference between the actual and asymptotic distributions in terms of the double-log scale. The error committed when replacing the actual distribution of the maximum under power normalization by its asymptotic distribution is studied, assuming that the cumulative distribution function of the random variables is known. Finally, we show by examples that the convergence to the asymptotic distribution may not be uniform in this double-log scale.