  Within the mode-coupling theory for structural relaxation in simple systems
the asymptotic laws and their leading-asymptotic correction formulas are
derived for the motion of a tagged particle near a glass-transition
singularity. These analytic results are compared with numerical ones of the
equations of motion evaluated for a tagged hard sphere moving in a hard-sphere
system. It is found that the long-time part of the two-step relaxation process
for the mean-squared displacement can be characterized by the $\alpha
$-relaxation-scaling law and von Schweidler's power-law decay while the
critical-decay regime is dominated by the corrections to the leading power-law
behavior. For parameters of interest for the interpretations of experimental
data, the corrections to the leading asymptotic laws for the non-Gaussian
parameter are found to be so large that the leading asymptotic results are
altered qualitatively by the corrections. Results for the non-Gaussian
parameter are shown to follow qualitatively the findings reported in the
molecular-dynamics-simulations work by Kob and Andersen [Phys. Rev. E 51, 4626
(1995)].
