Lévy based isotropic random fields on spheres

Hansen et al. (2011) introduced Lévy based models for three dimensional star-shaped random particles. The radial function of a Lévy particle arises from a kernel smoothing of a Lévy basis. The associated isotropic random field on the sphere has a correlation function which is given by the spherical self-convolution of the kernel. Using Schoenberg's (1942) characterisation of isotropic positive definite functions on spheres, we show that the approach of Hansen et al. (2011) is indeed very flexible: Any isotropic positive definite function on the sphere has a spherical convolution root. Some interesting consequences of this result will be discussed.

References:
Hansen, L. V., Thorarinsdottir, T. L., and Gneiting, T. (2011). Lévy particles: Modelling and simulating star-shaped random sets. CSGB Research Report.

Schoenberg, I. J. (1942). Positive definite functions on spheres. Duke Math. J., 9, 96-108.