On[alpha]-Symmetric Multivariate Characteristic Functions

Ann-dimensional random vector is said to have an[alpha]-symmetric distribution,[alpha]>0, if its characteristic function is of the form[phi]((u1[alpha]+...+un[alpha])1/[alpha]). We study the classes[Phi]n([alpha]) of all admissible functions[phi]: [0, [infinity])-->. It is known that members of[Phi]n(2) and[Phi]n(1) are scale mixtures of certain primitives[Omega]nand[omega]n, respectively, and we show that[omega]nis obtained from[Omega]2n-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: If[phi](0)=1,[phi]is continuous, limt-->[infinity] [phi](t)=0, and[phi](2n-2)(t) is convex, then[phi][set membership, variant][Phi]n(1). The paper closes with various criteria for the unimodality of an[alpha]-symmetric distribution.