Onα-Symmetric Multivariate Characteristic Functions

Abstract

Ann-dimensional random vector is said to have anα-symmetric distribution,α0, if its characteristic function is of the formφ((u1α+...+unα)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 Askey's 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. © 1998 Academic Press.