Basic Results of Extreme Value Theory

The development of probabilistic extreme value theory started with papers by Fisher and Tippett 1928, von Mises 1936 and Gnedenko 1943. Fisher and Tippet obtained three possible forms of the limiting distributions of maxima, called extreme value distributions, in the sequences of independent and identically distributed random variables. Von Mises obtained the sufficient conditions for an absolutely continuous distribution function to belong to any of the three possible domains of attraction for maxima. Gnedenko obtained the necessary and sufficient conditions for an arbitrary distribution function to belong to any of the domains of attraction. Extreme value theory was extended to stationary random sequences primarily owing to the paper by Leadbetter 1974. Extreme order statistics, multivariate extremes, extremes in incomplete samples from stationary sequences, extreme values in non-stationary sequences, extremes in random fields, and other topics have also attracted considerable attention. Although the statistical inference of extreme values is not in the focus of this book, let us mention here that a great deal of research has been devoted to the statistics of extreme values.