In certain cases the distribution of the normalized maximumof a sample can be better approximated by a sequence ofdifferent extreme value distributions than by the final one. Weshow that these cases are rather restricted and that the possibleimprovement is not spectacular.

The theory of stable probability distributions and their domains of attraction is derived in a direct way(avoiding the usual route via infinitely divisible distributions) using Fourier transforms. Regularly varyingfunctions play an important role in the exposition.

Consider n i.i.d. random vectors on R2, with unknown, common distribution function F. Under a sharpening of the extreme value condition on F, we derive a weighted approximation of the corresponding tail copula process. Then we construct a test to check whether the extreme value condition holds...