# Fractional extreme distributions

## TL;DR

This paper introduces fractional order differential equations for extreme value distributions, providing unique solutions expressed via stable subordinators and special functions, extending classical extreme value theory.

## Contribution

It develops a fractional generalization of classical extreme distributions, linking them to special functions and stochastic processes, and explores their analytical properties.

## Key findings

- Unique solutions expressed through exponential and stable subordinators.
- Connection with Kilbas-Saigo and Le Roy functions for different cases.
- Extended properties of classical Mittag-Leffler functions, including monotonicity and infinite divisibility.

## 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.