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

TL;DR

This paper investigates the self-decomposability and infinite divisibility of Fréchet and related distributions, revealing new conditions and interpretations through connections with Gamma subordinators and Lévy processes.

Contribution

It establishes that size-biased Fréchet distributions are self-decomposable and characterizes when the extreme value distribution is infinitely divisible, solving longstanding open problems.

Findings

Size-biased Fréchet distributions are self-decomposable.

Extreme value distribution is infinitely divisible iff ξ not in (0,1).

Provides analytical and probabilistic interpretations of infinite divisibility.

Abstract

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 $t,\alpha > 0.$