  The distribution of the initial short-time displacements of particles is
considered for a class of classical systems under rather general conditions on
the dynamics and with Gaussian initial velocity distributions, while the
positions could have an arbitrary distribution. This class of systems contains
canonical equilibrium of a Hamiltonian system as a special case. We prove that
for this class of systems the nth order cumulants of the initial short-time
displacements behave as the 2n-th power of time for all n>2, rather than
exhibiting an nth power scaling. This has direct applications to the initial
short-time behavior of the Van Hove self-correlation function, to its
non-equilibrium generalizations the Green's functions for mass transport, and
to the non-Gaussian parameters used in supercooled liquids and glasses.
