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Infinite divisibility of $α$-Cauchy and related variables

Min Wang

TL;DR

The paper resolves the infinite divisibility question for the $\alpha$-Cauchy distribution $\mathcal{C}_\alpha$ with $\alpha>1$ in a new regime $1<\alpha\le 6/5$ and provides sufficient conditions for the ID of powers $|\mathcal{C}_\alpha|^p$, addressing open questions by Yano, Yano and Yor. It develops a Gamma-type moment framework built around the random variables $X_{a,b,c,d}$ and leverages a three-parameter Mittag-Leffler function to translate existence of such moments into ID properties, using Kristiansen's theorem and $\mathbf{\Gamma}_2$-mixtures to obtain ID results for the original variable and its half-powers. The work yields three main contributions: (i) a positive ID result for $\mathcal{C}_\alpha$ in a nontrivial range, (ii) explicit sufficient conditions for the ID of $|\mathcal{C}_\alpha|^{\varepsilon p}$ across parameter regimes, and (iii) a broader Gamma-type existence theory with applications to half-stable and half Student-t distributions and to the nonnegativity of the three-parameter Mittag-Leffler function. These results advance understanding of ID for heavy-tailed distributions and provide new tools for analyzing related stochastic processes and special functions.

Abstract

We study the infinite divisibility of the $α$-Cauchy variable $\mathcal{C}_α$, $α> 1$. The distribution of $\mathcal{C}_2$ is the well-known Cauchy distribution, which is infinitely divisible and even stable. But when $α\neq 2$, there is no known result on the infinite divisibility of $\mathcal{C}_α$. In this paper, we prove that $\mathcal{C}_α$ is infinitely divisible if $1 < α\leq 6/5$, and we give some sufficient conditions for $|\mathcal{C}_α|^p, \, p\in \mathbb{R},$ to be infinitely divisible, which partially answers the open questions raised by Yano, Yano and Yor in 2009. In the proofs, a class of positive random variables having moments of Gamma type plays an important role, and we investigate the conditions for their existence.

Infinite divisibility of $α$-Cauchy and related variables

TL;DR

The paper resolves the infinite divisibility question for the -Cauchy distribution with in a new regime and provides sufficient conditions for the ID of powers , addressing open questions by Yano, Yano and Yor. It develops a Gamma-type moment framework built around the random variables and leverages a three-parameter Mittag-Leffler function to translate existence of such moments into ID properties, using Kristiansen's theorem and -mixtures to obtain ID results for the original variable and its half-powers. The work yields three main contributions: (i) a positive ID result for in a nontrivial range, (ii) explicit sufficient conditions for the ID of across parameter regimes, and (iii) a broader Gamma-type existence theory with applications to half-stable and half Student-t distributions and to the nonnegativity of the three-parameter Mittag-Leffler function. These results advance understanding of ID for heavy-tailed distributions and provide new tools for analyzing related stochastic processes and special functions.

Abstract

We study the infinite divisibility of the -Cauchy variable , . The distribution of is the well-known Cauchy distribution, which is infinitely divisible and even stable. But when , there is no known result on the infinite divisibility of . In this paper, we prove that is infinitely divisible if , and we give some sufficient conditions for to be infinitely divisible, which partially answers the open questions raised by Yano, Yano and Yor in 2009. In the proofs, a class of positive random variables having moments of Gamma type plays an important role, and we investigate the conditions for their existence.
Paper Structure (19 sections, 13 theorems, 60 equations)

This paper contains 19 sections, 13 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Infinite divisibility of -Cauchy variables
  3. Infinite divisibility of powers of half -Cauchy variable
  4. Sketch of the proofs
  5. Preliminaries
  6. Three-parametric Mittag-Leffler function
  7. Some criteria for determining infinite divisibility
  8. Some new moments of Gamma type
  9. Proof of Theorem \ref{['thm Gamma type']}
  10. Proof of Proposition \ref{['proposition main 1']}
  11. Proof of Proposition \ref{['proposition main 2']}
  12. Proof of Proposition \ref{['proposition main 3']}
  13. Proofs of Theorems \ref{['Theorem ID of alpha Cauchy']} and \ref{['Theorem ID of half alpha Cauchy']}
  14. Proof of Theorem \ref{['Theorem ID of alpha Cauchy']}
  15. Proof of Theorem \ref{['Theorem ID of half alpha Cauchy']}
  16. ...and 4 more sections

Key Result

Theorem 1.1

The $\alpha$-Cauchy variable $\mathcal{C}_\alpha$ is infinitely divisible if $1 < \alpha \leq 6/5$.

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Proposition 3.1
  • Theorem 3.3
  • Remark 3.4
  • Proposition 3.5
  • Proposition 3.6
  • ...and 6 more