Fractional extreme distributions

Abstract

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.