Isotropic Multiple Scattering Processes on Hyperspheres

TL;DR

This paper investigates isotropic random walks and multiple scattering processes on hyperspheres, deriving Fourier expansions, unimodality results, and asymptotic distributions, with applications to parameter estimation in Compound Cox Processes.

Contribution

It introduces new Fourier expansion techniques and asymptotic results for scattering processes on hyperspheres, especially involving von Mises Fisher distributions.

Findings

Fourier expansions for processes on hyperspheres derived.

Unimodality of multiconvolution of symmetric pdfs established.

Asymptotic distributions for vMF convolutions obtained.

Abstract

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 (pdf) 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 allows us to compute estimation bounds for the parameters of Compound Cox Processes (CCP) on ${\mathbb S}^{p-1}$.