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

Min Wang

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.

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

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 -Cauchy variable. In addition, we find that a random variable having moment of Gamma type if and only if is quasi infinitely divisible. From this perspective, we can solve many Hausdorff moment problems of sequences of factorial ratios.

Paper Structure

This paper contains 14 sections, 10 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Moments of Gamma type
  3. Three-parametric Mittag-Leffler function
  4. Powers of half -Cauchy variable
  5. Previous results on moments of Gamma type
  6. Some special cases
  7. Quasi-infinite divisibility
  8. Connections to Hausdorff moment problem
  9. Proofs
  10. Proof of Theorem \ref{['thm main']}
  11. Proof of proposition \ref{['proposition main 1']}
  12. Proof of proposition \ref{['proposition main 2']}
  13. Proof of proposition \ref{['proposition main 3']}
  14. Proof of Theorems \ref{['Theorem ID of half alpha Cauchy']}

Key Result

Proposition 1.1

Jan10 If ${\bf X}$ exists, then either $\gamma > 0$, or $\gamma = 0$ and $\delta \leq 0$.

Theorems & Definitions (12)

  • Proposition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Theorem 1.5: Kristiansen Kri94
  • Theorem 1.6
  • Proposition 3.1
  • Proposition 4.1
  • Proposition 4.2
  • Proposition 4.3
  • ...and 2 more