Let \(P_ t\) be the semigroup of the asymmetric simple exclusion process on \({\mathbb{Z}}\) evolving in the following way: \[ \eta \to \eta^{x,x+1},\quad at\quad rate\quad p\eta (x)(1-\eta (z)); \]
\[ \eta \to \eta^{x,x+1},\quad at\quad rate\quad q\eta (x)(1-\eta (x)), \] where \(q<p<1\) and \(p+q=1\). Let \(\nu_{\rho}\) be the Bernoulli measure of parameter \(\rho\) and take as initial distribution the product measure \({\bar \nu}{}_{\rho}\) whose marginals are \({\bar \nu}{}_{\rho}\{\eta (x)=1\}=0\) for \(x<0\), \(=1\) for \(x=0\), \(=\nu_{\rho}\{\eta (x)=1\}\) for \(x>0\). Set \(u_{\rho}=(p-q)(1-\rho)\) and denote by \(r_ x\) the space shift by x to the left as an operator acting on the probability measure on \(\{0,1\}^ Z\). Finally, let \(\lambda (r,t)=P[B_ t>r],\) where \(B_ t\) is the Brownian motion with diffusion coefficient \(u_{\rho}\). The authors prove that for all \(r\in R\), \[ \lim_{\epsilon \to 0}r_{u_{\rho}\epsilon^{-1}t+r\epsilon^{-1/2}}{\bar \nu}_{\rho}P_{\epsilon^{-1}t}=\lambda (r,t)\nu_ 0+(1-\lambda (r,t))\nu_{\rho}. \] Moreover, let \(X_ t\) be the position of the leftmost particle at time t, then the following central limit theorem holds: \((X_ t-u_{\rho}t)/\sqrt{t}\) converges to the distribution of \(B_ 1\).