ON ALPHA-SYMMETRICAL MULTIVARIATE CHARACTERISTIC FUNCTIONS

An n-dimensional random vector is said to have an alpha-symmetric dist ribution, alpha>0, if its characteristic function is of the form phi(( \u(1)\(proportional to)+...+\u(n)\(alpha))(1/alpha)). We study the cla sses Phi(n)(alpha) of all admissible functions phi: [0, infinity) --> R. II is known that members of Phi(n)(2) and Phi(n)(1) are scale mixtu res of certain primitives Omega(n) and omega(n), respectively, and we show that omega(n) is obtained from Omega(2n-1) by n - 1 successive in tegrations. Consequently, curious relations between 1- and 2- (or sphe rically) symmetric distributions arise. An analogue of Askey's criteri on gives a partial solution to a question of D. Sr. P. Richards: If ph i(0) = 1, phi is continuous, lim(t-->infinity) phi(t) = 0, and phi((2n -2))(t) is convex, then phi is an element of Phi(n)(1). The paper clos es with various criteria for the unimodality of an alpha-symmetric dis tribution. (C) 1998 Academic Press.