On the Density arising from the Domain of Attraction between Sum and Supremum: the $\alpha$-Sun operator

Abstract

We explore the analytic properties of the density function $h(x;\gamma ,\alpha )$, $x\in (0,\infty )$, $\gamma >0$, $0<\alpha <1$ which arises from the domain of attraction problem for a statistic interpolating between the supremum and sum of random variables. The parameter $\alpha$ controls the interpolation between these two cases, while $\gamma$ parametrises the type of extreme value distribution from which the underlying random variables are drawn from. For $\alpha =0$ the Fréchet density applies, whereas for $\alpha =1$ we identify a particular Fox H-function, which are a natural extension of hypergeometric functions into the realm of fractional calculus. In contrast for intermediate $\alpha$ an entirely new function appears, which is not one of the extensions to the hypergeometric function considered to date. We derive series, integral and continued fraction representations of this latter function.

31 pages, 6 figures

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