Onα-Symmetric Multivariate Characteristic Functions

Ann-dimensional random vector is said to have anα-symmetric distribution,α>0, if its characteristic function is of the formϕ((|u1|α+…+|un|α)1/α). We study the classesΦn(α) of all admissible functionsϕ: [0, ∞)→R. It is known that members ofΦn(2) andΦn(1) are scale mixtures of certain primitivesΩnandωn, respectively, and we show thatωnis obtained fromΩ2n−1byn−1 successive integrations. Consequently, curious relations between 1- and 2- (or spherically) symmetric distributions arise. An analogue of Askeyʹs criterion gives a partial solution to a question of D. St. P. Richards: Ifϕ(0)=1,ϕis continuous, limt→∞ ϕ(t)=0, andϕ(2n−2)(t) is convex, thenϕ∈Φn(1). The paper closes with various criteria for the unimodality of anα-symmetric distribution.