On the self-decomposability of the Fréchet distribution

Let {Γt,t≥0} be the Gamma process. Using a moment identification due to Bertoin and Yor (2002)  [4] we observe that for every t>0t>0 and α∈(0,1)α∈(0,1) the random variable Γt−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 FξFξ is infinitely divisible if and only if ξ∉(0,1)ξ∉(0,1), solving problems raised by Steutel (1973)  [19] and Bondesson (1992)  [6]. We also review different analytical and probabilistic interpretations of the infinite divisibility of Γt−α for t,α>0t,α>0.