Approximation of multivariate stable densities

Multivariate stable densities do not generally have explicit formulas. We define approximate factorization for a general distribution and show that it is equivalent to independence for symmetric stable distributions. We define uniformly round distributions and show that it leads to continuous spectral measure. We introduce a new formula for multivariate symmetric $\alpha$-stable densities ($S\alpha S$) that only requires knowing the characteristic function on the unit sphere (a compact set), plus a function $g\sb{\alpha,d}$ that is the same for every $S\alpha S$ distribution. Furthermore, this formula provides more information about the density. We approximate the pdf at and in a neighborhood of the origin in terms of the decay of the characteristic function. In one dimension, we dominate one stable density by a multiple of another everywhere on $\IR$ while in multidimensions we dominate and compare stable densities on bounded regions. We give numerical calculation of a function g used in the new formula for multivariate stable densities and of densities for multivariate radially symmetric stable distributions. Finally, we give some applications.