Positive definite symmetric functions on finite dimensional spaces. I. Applications of the Radon transform

AbstractAn n-dimensional random vector X is said (Cambanis, S., Keener, R., and Simons, G. (1983). J. Multivar. Anal., 13 213–233) to have an α-symmetric distribution, α > 0, if its characteristic function is of the form φ(|ξ1|α + … + |ξn|α). Using the Radon transform, integral representations are obtained for the density functions of certain absolutely continuous α-symmetric distributions. Series expansions are obtained for a class of apparently new special functions which are encountered during this study. The Radon transform is also applied to obtain the densities of certain radially symmetric stable distributions on Rn. A new class of “zonally” symmetric stable laws on Rn is defined, and series expansions are derived for their characteristic functions and densities