Multivariate Stable Densities as Functions of One Dimensional Projections

The density of a generald-dimensional stable random vectorXis expressed as an integral over the sphere in Rdof a function of the parameters of the one dimensional projections ofX. These formulas give insight into the form of multivariate stable densities and are useful for numerical calculations. Corollaries give simplified expressions for symmetric stable and theα=1 strictly stable densities, relations among the densities in different dimensions, and values of the densities at the location parameter for all cases except theα=1, non-strictly stable ones. Expressions for the densities in the multidimensional analog of Zolotarevʹs (M) parameterization and a discussion of computational versions of the formulas are also given.