Cosmology based on large scale peculiar velocity prefers volume weighted velocity statistics. However, measuring the volume weighted velocity statistics from inhomogeneously distributed galaxies (simulation particles/halos) suffers from an inevitable and significant sampling artifact. We study this sampling artifact in the velocity power spectrum measured by the nearest particle velocity assignment method by Zheng et al., [Phys. Rev. D 88, 103510 (2013).]. 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 $k=0.1h/\mathrm{Mpc}$, for samples with number density ${10}^{-3}(\mathrm{Mpc}/h{)}^{-3}$. 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. A more robust evaluation of related statistics through simulations will be presented in a companion paper by Zheng et al., [Sampling artifact in volume weighted velocity measurement. II. Detection in simulations and comparison with theoretical modelling, arXiv:1409.6809.]. 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.

• Received 30 May 2014

DOI:https://doi.org/10.1103/PhysRevD.91.043522