  We investigate the velocity-correlation distributions after $n$ collisions of
a tagged particle undergoing binary collisions. Analytical expressions are
obtained in any dimension for the velocity-correlation distribution after the
first-collision as well as for the velocity-correlation function after an
infinite number of collisions, in the limit of Gaussian velocity distributions.
It appears that the decay of the first-collision velocity-correlation
distribution for negative argument is exponential in any dimension with a
coefficient that depends on the mass and on the coefficient of restitution. We
also obtained the velocity-correlation distribution when the velocity
distributions are not Gaussian: by inserting Sonine corrections of the velocity
distributions, we derive the corrections to the velocity-correlation
distribution which agree perfectly with a DSMC (Direct Simulation Monte Carlo)
simulation. We emphasize that these new quantities can be easily obtained in
simulations and likely in experiments: they could be an efficient probe of the
local environment and of the degree of inelasticity of the collisions.
