Asymptotics; Stable Laws for Extremes; Tail Properties

When for a set of random variables the underlying univariate and multivariate distributions are only approximately known, they become useless even for estimating the values of the distribution function of the extremes. In some exact models, computational difficulties might arise. Both of these can be overcome in asymptotic models. The paper discusses asymptotic models with emphasis on the availability of several dependent extreme value models. For the classical model of independent and identically distributed random variables, a functional equation is deduced for the possibly asymptotic (stable) distributions of normalized extremes. The solution of the functional equation is discussed, with emphasizing the necessity of assumptions on the domain of the equation in order to obtain the classical theory of extremes. Extensions to random sample sizes, and other related characterization theorems are also mentioned.

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