  We prove a Functional Central Limit Theorem for the position of a Tagged
Particle in the one-dimensional Asymmetric Simple Exclusion Process in the
hyperbolic scaling, starting from a Bernoulli product measure conditioned to
have a particle at the origin. We also prove that the position of the Tagged
Particle at time $t$ depends on the initial configuration, by the number of
empty sites in the interval $[0,(p-q)\alpha t]$ divided by $\alpha$ in the
hyperbolic and in a longer time scale, namely $N^{4/3}$.
