  The peculiar velocities of biased tracers of the cosmic density field contain
important information about the growth of large scale structure and generate
anisotropy in the observed clustering of galaxies. Using N-body data, we show
that velocity expansions for halo redshift-space power spectra are converged at
the percent-level at perturbative scales for most line-of-sight angles $\mu$
when the first three pairwise velocity moments are included, and that the third
moment is well-approximated by a counterterm-like contribution. We compute
these pairwise-velocity statistics in Fourier space using both Eulerian and
Lagrangian one-loop perturbation theory using a cubic bias scheme and a
complete set of counterterms and stochastic contributions. We compare the
models and show that our models fit both real-space velocity statistics and
redshift-space power spectra for both halos and a mock sample of galaxies at
sub-percent level on perturbative scales using consistent sets of parameters,
making them appealing choices for the upcoming era of spectroscopic,
peculiar-velocity and kSZ surveys.
