Sampling Artifact in Volume Weighted Velocity Measurement.--- I.
  Theoretical Modelling
  Cosmology based on large scale peculiar velocity preferes volume weighted
velocity statistics. However, measuring the volume weighted velocity statistics
from inhomogeneously distributed galaxies (simulation particles/halos) suffer
from an inevitable and significant sampling artifact. We study this sampling
artifact in the velocity power spectrum measured by the nearest-particle (NP)
velocity assignment method(Zheng et al. 2013, PRD). We derive the analytical
expression of leading and higher order terms. We find that the sampling
artifact suppresses the $z=0$ E-mode velocity power spectrum by $\sim 10\%$ at
$k=0.1h/$Mpc , for samples with number density $10^{-3}({\rm Mpc}/h)^{-3}$.
This suppression becomes larger for larger $k$ and for sparser samples. We
argue that, this source of systematic errors in peculiar velocity cosmology,
albeit severe, can be self-calibrated in the framework of our theoretical
modelling. We also work out the sampling artifact in the density-velocity cross
power spectrum measurement. More robust evaluation of related statistics
through simulations will be presented in a companion paper (Zheng, Zhang &
Jing, 2015, PRD). We also argue that similar sampling artifact exists in other
velocity assignment methods and hence must be carefully corrected to avoid
systematic bias in peculiar velocity cosmology.
