关于Fréchet分布的自分解性
On the self-decomposability of the Fréchet distribution
摘要：
设$\{Γ_t, \, t\ge 0\}$为Gamma从属子。利用Bertoin-Yor（2002）的矩识别，我们观察到对于每个$t > 0$和$α\in (0,1)$，随机变量$Γ_t^{-α}$分布为某个谱负Lévy过程的指数泛函.这意味着Fréchet分布的所有大小偏置样本都是自分解的，并且极值分布$F_ξ$是无限可分的当且仅当$ξ\not\in (0,1),$解决了Steutel（1973）和Bondesson（1992）提出的问题。我们还回顾了$t,α> 0.$的$Γ_t^{-α}$的无限可分性的不同分析和概率解释
英文摘要：
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 $t,α> 0.$
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