A note on the $\alpha-$Sun distribution

TL;DR

This paper studies the mathematical properties of the alpha-Sun distribution, revealing its infinite divisibility in the Fréchet case and detailed density behaviors at zero and infinity, thus addressing open questions in prior research.

Contribution

It provides new analytical insights into the alpha-Sun distribution, including density behaviors and infinite divisibility, expanding understanding of its properties in different cases.

Findings

Alpha-Sun distribution is infinitely divisible in the Fréchet case.

Exact behavior of the density at zero and infinity is characterized.

Answers open questions from Witte and Greenwood (2020).

Abstract

We investigate the analytical properties of the $\alpha-$Sun random variable, which arises from the domain of attraction of certain storage models involving a maximum and a sum. In the Fr\'echet case we show that this random variable is infinitely divisible, and we give the exact behaviour of the density at zero. In the Weibull case we give the exact behaviour of the density at infinity, and we show that the behaviour at zero is neither polynomial nor exponential. This answers the open questions in the recent paper Witte and Greenwood (2020).