| Abstract: | If suitably normalized maxima of an i.i.d. sample converge in distribution, the limiting distribution is known to be max-infinitely divisible and the common distribution of the sample is said to belong to its domain of attraction. We prove the existence of max-universal distributions belonging to the domain of attraction of every max-infinitely divisible law. The proof follows in the spirit of corresponding results for normalized sums of i.i.d. random variables originated by Doeblin and shows that necessarily the sampling size has to be rapidly increasing. Restricting the growth rate of the sampling size, we prove that one necessarily deals with max-semistable distributions and their domains of attraction. 2000 Mathematics subject classification Primary—60G70 Secondary—60E99, 60F05 |
