Central limit theorem for a tagged particle in asymmetric simple exclusion

We prove a functional central limit theorem for the position of a tagged particle in the one-dimensional asymmetric simple exclusion process for 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$t$ depends on the initial configuration, through the number of empty sites in the interval [0,(p−q)αt]$[0,(p-q)\alpha t]$ divided by α$\alpha$, on the hyperbolic time scale and on a longer time scale, namely N^4/3$N^{4/3}$.