Extreme values Nonlinear normalization Limit theorems Domain of attraction
Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Pliska Studia Mathematica Bulgarica, Vol. 20, No 1, (2011), 169p-186p
It is well known that under linear normalization the maxima of iid random variables converges in distribution to one of the three types of max-stable laws: Frechet, Gumbel and Weibull. During the last two decades the first author and her collaborators worked out a limit theory for extremes and extremal processes under non-linear but monotone normalizing mappings. In this model there is only one type of max-stable distributions and all continuous and strictly increasing df's belong to it. In a recent paper on General max-stable laws, Sreehari points out two "confusing" results in Pancheva (1984). They concern the explicit form of a max-stable df with respect to a continuous one-parameter group of max-automorphisms, and domain of attraction conditions. In the present paper the first claim is answered by a detailed explanation of the explicit form, while for the second we give a revised proof. The rate of convergence is also discussed.