2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/122515We 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 $[0,(p-q)αt]$ divided by $α$ in the hyperbolic and in a longer time scale, namely $N^{4/3}$.28 pages, no figuresProbability60K35text