Infinite divisibility of $\alpha$-Cauchy distributions

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 $1 < \alpha \leq 2$.