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 law is in one-to-one correspondence with a Kilbas-Saigo function for the Weibull and Fr\'echet cases, and with a Le Roy function for the Gumbel case. By the stochastic representation, we can derive several analytical properties for the latter special functions, extending known features of the classical Mittag-Leffler function, and dealing with monotonicity, complete monotonicity, infinite divisibility, asymptotic behaviour at infinity, uniform hyperbolic bounds.