A Probabilistic Perspective on Feller, Pollard and the Complete Monotonicity of the Mittag-Leffler Function

TL;DR

This paper employs probability theory to demonstrate that the three-parameter Mittag-Leffler function is completely monotone by representing it as a Laplace transform of a distribution, extending previous methods and providing new integral representations.

Contribution

It introduces a probabilistic approach to establish complete monotonicity of the three-parameter Mittag-Leffler function, expanding beyond prior contour and Laplace transform techniques.

Findings

Proves the three-parameter Mittag-Leffler function is the Laplace transform of a distribution.

Provides a new integral representation of the Prabhakar function.

Extends probabilistic methods to analyze complete monotonicity.

Abstract

The main contribution of this paper is the use of probability theory to prove that the three-parameter Mittag-Leffler function is the Laplace transform of a distribution and thus completely monotone. Pollard used contour integration to prove the result in the one-parameter case. He also cited personal communication by Feller of a discovery of the result by ''methods of probability theory''. Feller used the two-dimensional Laplace transform of a bivariate distribution to derive the result. We pursue the theme of probability theory to explore complete monotonicity beyond the contribution due to Feller. Our approach involves an interplay between mixtures and convolutions of stable and gamma densities, together with a limit theorem that leads to a novel integral representation of the three-parameter Mittag-Leffler function (also known as the Prabhakar function).