Generalized Gamma convolutions and complete monotonicity

Abstract

Let E be the class of pdf's f on (0,∞) such that, for each u>0, f(uv) f (u/v) is completely monotone as a function of w=v+v-1. This class includes many familiar pdf's and is closed with respect to multiplication and division of independent rv's. Further, E ⊂T, where T is the class of generalized Gamma convolutions (GGC) introduced by O. Thorin. Moreover, E coincides with the class of pdf's of the form {Mathematical expression} (all parameters positive) or limits thereof. The Laplace transform φ{symbol} of a GGC is characterized by complete monotonicity of φ{symbol}(uv) φ{symbol}(u/v) as a function of w. This characterization has many consequences and applications. It follows that also the class T has simple multiplicative properties. © 1990 Springer-Verlag.