Infinite divisibility of powers of positive stable random variables

Source abstract

Let $Z_α$ be a positive $α$-stable random variable, where $0 0$. Combining this result with Jedidi and Simon's characterization of infinite divisibility of negative powers of $Z_α$, we obtain, for $p\ne0$, $$Z_α^p\ \text{is infinitely divisible} \quad\Longleftrightarrow\quad p\in\left(-\infty,-\fracα{1-α}\right]\cup(0,\infty).$$ The proof relies on Kanter's factorization and a general product property: if $X$ is a mixture of exponential distributions, $W$ is a generalized gamma convolution, and $X$ and $W$ are independent, then $(X+c)W$ is infinitely divisible for every $c\ge 0$.