Determinantal point processes on spheres: multivariate linear statistics

TL;DR

This paper derives Wiener chaos decompositions and cumulant representations for multivariate linear statistics of determinantal point processes on spheres, providing precise spectral kernel estimates and symmetry-based identities.

Contribution

It introduces the first and second order Wiener chaos decompositions for these statistics on spheres, with new graphical cumulant representations and spectral kernel analysis.

Findings

Wiener chaos decompositions for multivariate statistics on spheres

Graphical cumulant representations for determinantal point processes

Precise spectral kernel estimates and identities

Abstract

In this paper, we will derive the first and 2nd order Wiener chaos decomposition for the multivariate linear statistics of the determinantal point processes associated with the spectral projection kernels on the unit spheres $S^d$. We will first get a graphical representation for the cumulants of multivariate linear statistics for any determinantal point process. The main results then follow from the very precise estimates and identities regarding the spectral projection kernels and the symmetry of the spheres.