Functions operating on multivariate distribution and survival functions—With applications to classical mean-values and to copulas

Abstract

Functions operating on multivariate distribution and survival functions are characterized, based on a theorem of Morillas, for which a new proof is presented. These results are applied to determine those classical mean values on [ 0 , 1 ] n which are distribution functions of probability measures on [ 0 , 1 ] n . As it turns out, the arithmetic mean plays a universal rôle for the characterization of distribution as well as survival functions. Another consequence is a far reaching generalization of Kimberling’s theorem, tightly connected to Archimedean copulas.