Multivariate Stable Densities as Functions of One Dimensional Projections

Abstract

The density of a general d-dimensional stable random vector X is expressed as an integral over the sphere in R d of a function of the parameters of the one dimensional projections of X. 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 the :=1 strictly stable densities, relations among the densities in different dimensions, and values of the densities at the location parameter for all cases except the :=1, non-strictly stable ones. Expressions for the densities in the multidimensional analog of Zolotarev's (M) parameterization and a discussion of computational versions of the formulas are also given. Academic Press

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