Open Access
2020 Fractional extreme distributions
Lotfi Boudabsa, Thomas Simon, Pierre Vallois
Electron. J. Probab. 25: 1-20 (2020). DOI: 10.1214/20-EJP520


We consider three classes of linear differential equations on distribution functions, with a fractional order $\alpha \in [0,1]$. The integer case $\alpha =1$ corresponds to the three classical extreme families. In general, we show that there is a unique distribution function solving these equations, whose underlying random variable is expressed in terms of an exponential random variable and an integral transform of an independent $\alpha $-stable subordinator. From the analytical viewpoint, this distribution is in one-to-one correspondence with a Kilbas-Saigo function for the Weibull and Fréchet cases, and with a Le Roy function for the Gumbel case.


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Lotfi Boudabsa. Thomas Simon. Pierre Vallois. "Fractional extreme distributions." Electron. J. Probab. 25 1 - 20, 2020.


Received: 31 July 2019; Accepted: 5 September 2020; Published: 2020
First available in Project Euclid: 25 September 2020

MathSciNet: MR4161125
Digital Object Identifier: 10.1214/20-EJP520

Primary: 26A33 , 33E12 , 45E10 , 60E05 , 60G52

Keywords: double Gamma function , extreme distribution , fractional differential equation , Kilbas-Saigo function , Le Roy function , stable subordinator

Vol.25 • 2020
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