  The power spectrum and the voxel intensity distribution (VID) are two of the
main proposed summary statistics to study line-intensity maps. We reformulate
the derivation of the VID in terms of the local overdensities and derive for
the first time an analytic covariance between the VID and the line-intensity
mapping power spectrum. We study the features of this covariance for different
experimental setups and show that we can recover similar results to
simulation-based covariances. With this formalism, we also compute the cosmic
variance contribution the VID uncertainty, which we find to be subdominant with
respect to the standard variance from Poisson sampling. Our results allow for
general joint analyses of the VID and the line-intensity mapping power
spectrum.
