Infinite divisibility of $α$-Cauchy distributions

Abstract

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