Convolutions and Mixtures of Gamma, Stable and Mittag-Leffler Distributions

Abstract

This paper uses convolutions of the gamma density and the one-sided stable density to construct higher level densities. The approach is applied to constructing a 4-parameter Mittag-Leffler density, whose Laplace transform is a corresponding Mittag-Leffler function, which is completely monotone (CM) by construction. Laplace transforms of mixtures of the stable densities with respect to the 4-parameter Mittag-Leffler distribution are compositions of the Mittag-Leffler functions with Bernstein functions, thereby generating a rich family of CM variants of the base CM Mittag-Leffler functions, including known instances as special cases.