Homogeneous distributions—And a spectral representation of classical mean values and stable tail dependence functions

Homogeneous distributions on R+d and on R¯+d∖︀{∞¯d} are shown to be Bauer simplices when normalized. This is used to provide spectral representations for the classical power mean values Mt(x)Mt(x) which turn out to be unique mixtures of the functions x⟼mini≤d(aixi)x⟼mini≤d(aixi) for t≤1t≤1 (with some gaps depending on the dimension dd), resp. x⟼maxi≤d(aixi)x⟼maxi≤d(aixi) for t≥1t≥1 (without gaps), where ai≥0ai≥0.There exists a very close connection with so-called stable tail dependence functions of multivariate extreme value distributions. Their characterization by Hofmann (2009) [7] is improved by showing that it is not necessary to assume the triangle inequality — which instead can be deduced.