On the density arising from the domain of attraction of an operator interpolating between sum and supremum: The α-Sun operator

Abstract

We explore the analytic properties of the density function h(x;γ,α), x∈(0,∞), γ>0, 0<α<1 which arises as a normed limit from the domain of attraction problem for an operator interpolating between the supremum and sum as applied to a sequence of i.i.d. non-negative random variables. The parameter α controls the interpolation between these two cases, while γ further parametrises the type of distribution from which the underlying random variables are drawn. It is known that in the normed limit, for α=0 the Fréchet density arises, whereas for α=1 the limit is a stable random variable which has the well-known identification with a particular Fox H-function. It is known [21] that for intermediate α an entirely new distribution function appears, which is not one of the extensions to the hypergeometric function considered to date. Here we derive series, integral and continued fraction representations of this latter function.