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

TL;DR

This paper develops new methods for constructing complex probability densities using convolutions of gamma and stable distributions, leading to a family of Mittag-Leffler densities with applications in modeling and analysis.

Contribution

It introduces a novel convolution-based approach to build higher-level densities, including a 4-parameter Mittag-Leffler density with completely monotone properties.

Findings

Constructed a 4-parameter Mittag-Leffler density with CM properties

Generated a family of CM variants through mixtures involving Bernstein functions

Included known distributions as special cases within the new family

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.