On the self-decomposability of the Fréchet distribution

Abstract

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