  We present a simple model for the shape of the distribution function of
galaxy peculiar velocities. We show how both nonlinear and linear theory terms
combine to produce a distribution which has an approximately Gaussian core with
exponential wings. The model is easily extended to study how the statistic
depends on the type of particle used to trace the velocity field (dark matter
particles, dark matter haloes, galaxies), and on the density of the environment
in which the test particles are. Comparisons with simulations suggest that our
model is accurate. We also show that the evolution of the peculiar velocities
depends on the local, rather than the global density. Since clusters populate
denser regions on average, using cluster velocities with the linear theory
scaling may lead to an overestimate of the global value of Omega. Conversely,
using linear theory with the global value of Omega to scale cluster velocities
from the initial to the present time results in an underestimate of their true
velocities. In general, however, the directions of motions of haloes are rather
well described by linear theory. Our results help to simplify models of
redshift-space distortions considerably.
