On the self-decomposability of the Fr\'echet distribution

Let $\{\Gamma_t, \, t\ge 0\}$ be the Gamma subordinator. Using a moment identification due to Bertoin-Yor (2002), we observe that for every $t > 0$ and $\alpha\in (0,1)$ the random variable $\Gamma_t^{-\alpha}$ is distributed as the exponential functional of some spectrally negative L\'evy process. This entails that all size-biased samplings of Fr\'echet distributions are self-decomposable and that the extreme value distribution $F_\xi$ is infinitely divisible if and only if $\xi\not\in (0,1),$ solving problems raised by Steutel (1973) and Bondesson (1992). We also review different analytical and probabilistic interpretations of the infinite divisibility of $\Gamma_t^{-\alpha}$ for $t,\alpha > 0.$