This is the second submission of this document. Continuing the line of research initiated in Iksanov and Möhle (2008) and Negadajlov (2008) we investigate the asymptotic (as n→∞) behaviour of Vn the number of zero increments before the absorption in a random walk with the barrier n. In particular, when the step of the unrestricted random walk has a finite mean, we prove that the number of zero increments converges in distribution. We also establish a weak law of large numbers for Vn under a regular variation assumption.