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 M t ( x ) which turn out to be unique mixtures of the functions x ⟼ min i ≤ d ( a i x i ) for t ≤ 1 (with some gaps depending on the dimension d ), resp. x ⟼ max i ≤ d ( a i x i ) for t ≥ 1 (without gaps), where a i ≥ 0 . 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.