  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.
