Self-decomposability of $α$-Cauchy distributions

Source abstract

In 2009, Yano, Yano and Yor asked whether the $α$-Cauchy distributions are self-decomposable or at least infinitely divisible in their study of hitting times for symmetric stable processes. Recently, the first named author proved that $α$-Cauchy distribution is infinitely divisible if and only if $1 < α\leq 2$. In this paper, we prove that $α$-Cauchy distribution is self-decomposable if and only if $α= 2$. The proof is based on a new criterion for self-decomposability in the symmetric case. As an application, we show that the first hitting time of a nonzero point by a symmetric stable process of index $1<α<2$ starting from zero is not self-decomposable.