Moments of Gamma type and three-parametric Mittag-Leffler function

TL;DR

This paper explores a class of positive random variables with Gamma-type moments, linking them to three-parametric Mittag-Leffler functions, and investigates their properties, including infinite divisibility and moment problems.

Contribution

It introduces conditions for the existence and non-negativity of these functions and connects Gamma-type moments to quasi infinite divisibility, advancing understanding of related stochastic properties.

Findings

Conditions for existence of Gamma-type moments

Criteria for non-negativity of three-parametric Mittag-Leffler functions

Characterization of infinite divisibility of powers of auchy variables

Abstract

We study a class of positive random variables having moments of Gamma type, whose density can be expressed by the three-parametric Mittag-Leffler functions. We give some necessary conditions and some sufficient conditions for their existence. As a corollary, we give some conditions for non-negativity of the three-parametric Mittag-Leffler functions. As an application, we study the infinite divisibility of the powers of half $\a$-Cauchy variable. In addition, we find that a random variable $\X$ having moment of Gamma type if and only if $\log \X$ is quasi infinitely divisible. From this perspective, we can solve many Hausdorff moment problems of sequences of factorial ratios.