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 tt depends on the initial configuration, through the number of empty sites in the interval [0,(p−q)αt][0,(p−q)αt] divided by αα, on the hyperbolic time scale and on a longer time scale, namely N4/3N4/3.