Isotropic Multiple Scattering Processes on Hyperspheres

This paper presents several results about isotropic random walks and multiple scattering processes on hyperspheres ${\mathbb S}^{p-1}$ . It allows one to derive the Fourier expansions on ${\mathbb S}^{p-1}$ of these processes. A result of unimodality for the multiconvolution of symmetrical probability density functions on ${\mathbb S}^{p-1}$ is also introduced. Such processes are then studied in the case where the scattering distribution is von Mises–Fisher (vMF). Asymptotic distributions for the multiconvolution of vMFs on ${\mathbb S}^{p-1}$ are obtained. Both Fourier expansion and asymptotic approximation allow us to compute estimation bounds for the parameters of compound cox processes on ${\mathbb S}^{p-1}$ .