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We define a new class of positive and Lebesgue measurable functions in terms of their asymptotic behavior, which includes the class of regularly varying functions. We also characterize it by transformations, corresponding to generalized moments when these functions are random variables. We study the properties of this new class and discuss their applications to Extreme Value Theory.

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Very recently Seneta [15] has provided a characterization of slowly varying functions L in the Zygmund sense by using the condition, for each y > 0,We extend this result by considering a wider class of functions and a more general condition than (1). Further, a representation theorem for this wider class is provided.

This paper provides new properties for tails of probability distributions belonging to a class defined according to the asymptotic decay of the tails. This class contains the one of regularly varying tails of distributions. The main results concern the relation between this larger class and the maximum domains of attraction of Fréchet and Gumbel.

Recently, new classes of positive and measurable functions, M(ρ) and M(±∞), have been defined in terms of their asymptotic behaviour at infinity, when normalized by a logarithm (Cadena et al., , 2017. Looking for other suitable normalizing functions than logarithm seems quite natural. It is what is developed in this paper, studying new classes of functions of the type lim x→∞ log U (x)/H(x) = ρ < ∞ for a large class of normalizing functions H. It provides subclasses of M(0) and M(±∞).

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