Infinite divisibility of $\alpha$-Cauchy distributions

TL;DR

This paper investigates the infinite divisibility of $ ext{alpha}$-Cauchy distributions, establishing that they are infinitely divisible precisely when $1 < ext{alpha} extless 2$, extending understanding beyond the classical Cauchy case.

Contribution

The paper proves the exact range of $ ext{alpha}$ for which $ ext{alpha}$-Cauchy distributions are infinitely divisible, resolving a question posed in 2009.

Findings

$ ext{alpha}$-Cauchy is infinitely divisible if and only if $1 < ext{alpha} extless 2$

Standard Cauchy ($ ext{alpha}=2$) is confirmed to be infinitely divisible

For $ ext{alpha} eq 2$, infinite divisibility does not hold

Abstract

In 2009, Yano, Yano and Yor proposed the question of studying the infinite divisibility of the $\alpha$-Cauchy variable $\mathcal{C}_\alpha$ for $\alpha > 1$. The particular case $\mathcal{C}_2$ is the well-known standard Cauchy variable, which is infinitely divisible and indeed stable. For $\alpha \neq 2$, the infinite divisibility of $\mathcal{C}_\alpha$ is previously unknown. In this paper, we prove that $\mathcal{C}_\alpha$ is infinitely divisible if and only if $1 < \alpha \leq 2$.