Slow Variation and Characterization of Domains of Attraction

Abstract

Suppose G is the distribution function of an extreme-value distribution. We shall develop necessary and sufficient conditions for a distribution function F such that for some choice of constants a_n > 0 and b_n (n = 1, 2,…)

$${{\text{F}}^{\text{n}}}({{\text{a}}_{\text{n}}}{\text{x + }}{{\text{b}}_{\text{n}}}) \to {\text{G(x) weakly (n}} \to \infty {\text{)}}$$

(1)

The left hand side is the distribution function of the normalized maximum of n i.i.d. random variables with distribution function F.