  We present a numerical study of the properties of the stellar velocity
distribution in stellar discs which have developed a saturated, two-armed
spiral structure. We follow the growth of the spiral structure deeply into the
non-linear regime by solving the Boltzmann moment equations up to second order.
By adopting the thin-disc approximation, we restrict our study of the stellar
velocity distribution to the plane of the stellar disc. We find that the outer
(convex) edges of stellar spiral arms are characterized by peculiar properties
of the stellar velocity ellipsoids, which make them distinct from most other
galactic regions. In particular, the ratio \sigma_1:\sigma_2 of the smallest
versus largest principal axes of the stellar velocity ellipsoid can become
abnormally small (as compared to the rest of the disc) near the outer edges of
spiral arms. Moreover, the epicycle approximation fails to reproduce the ratio
of the tangential versus radial velocity dispersions in these regions. These
peculiar properties of the stellar velocity distribution are caused by
large-scale non-circular motions of stars, which in turn are triggered by the
non-axisymmetric gravitational field of stellar spiral arms. The magnitude of
the vertex deviation appears to correlate globally with the amplitude of the
spiral stellar density perturbations. However, locally there is no simple
correlation between the vertex deviation and the density perturbations.
(Abstract abridged).
