  The streaming model describes the mapping between real and redshift space for
2-point clustering statistics. Its key element is the probability density
function (PDF) of line-of-sight pairwise peculiar velocities. Following a
kinetic-theory approach, we derive the fundamental equations of the streaming
model for ordered and unordered pairs. In the first case, we recover the
classic equation while we demonstrate that modifications are necessary for
unordered pairs. We then discuss several statistical properties of the pairwise
velocities for DM particles and haloes by using a suite of high-resolution
$N$-body simulations. We test the often used Gaussian ansatz for the PDF of
pairwise velocities and discuss its limitations. Finally, we introduce a
mixture of Gaussians which is known in statistics as the generalised hyperbolic
distribution and show that it provides an accurate fit to the PDF. Once
inserted in the streaming equation, the fit yields an excellent description of
redshift-space correlations at all scales that vastly outperforms the Gaussian
and exponential approximations. Using a principal-component analysis, we reduce
the complexity of our model for large redshift-space separations. Our results
increase the robustness of studies of anisotropic galaxy clustering and are
useful for extending them towards smaller scales in order to test theories of
gravity and interacting dark-energy models.
