Which distributions in the max-domain of attraction satisfy von Mises representation or variation representation for a given auxiliary function?

Abstract In this paper we characterize the family of distributions F which are in the max-domain of attraction (MDA) of Gumbel, Fréchet, or Weibull extreme value distributions. The extreme value behavior of MDA distribution F can be analyzed in terms of two convenient representations: the von Mises representation ( $$\textsf {vMR}$$ ) and the variation representation ( $$\textsf {VR}$$ ), where the latter covers both regular and gamma variation. Each of these representations is determined by an appropriate auxiliary function, the choice of which is in focus of our analysis. In particular, our main result provides the necessary and sufficient conditions that an MDA distribution F allows $$\textsf {vMR}$$ or $$\textsf {VR}$$ for a given auxiliary function. Moreover, we identify the exact classes of MDA distributions F allowing $$\textsf {vMR}$$ (or $$\textsf {VR}$$ ) with the same auxiliary function $$\psi _{{\textsf {vMR}}}$$ (or $$\psi _{{\textsf {VR}}}$$ ). Hence, our findings complete the characterization of the relation between MDA distributions F and the corresponding $$\textsf {vMR}$$ or $$\textsf {VR}$$ auxiliary functions.