Determinantal point processes on spheres: multivariate linear statistics

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.