Spherical harmonic analysis for multivariate stable distributions

TL;DR

This paper develops spherical harmonic series representations for multivariate stable distributions, enabling explicit calculations and asymptotic analysis of their densities and characteristic exponents.

Contribution

It introduces new series representations using spherical harmonics for stable distributions, with convergence results and practical computation methods.

Findings

Series representations converge absolutely for certain stable distributions.

Explicit formulas for characteristic exponents and densities are derived.

Asymptotic expansions provide insights into tail behaviors.

Abstract

Series representations consisting of spherical harmonics are obtained for characteristic exponents and probability density functions of multivariate stable distributions under various conditions. A esult potentially applicable in a practical setting is that for any distribution with stability index not equal to 1 and with a polynomial spectral spherical density, the series representation converges absolutely with all terms being calculable in closed form. Asymptotic expansions consisting of spherical harmonics are also considered for probability density functions.