We continue the line of research of random walks with a barrier initiated by Iksanov and Möhle (2008). Assuming that the tail of the step of the underlying random walk has a power-like behavior at infinity with the exponent $-\alpha$, $\alpha\in.
"Weak convergence of the number of zero increments in the random walk with barrier." Electron. Commun. Probab. 19 1 - 11, 2014. https://doi.org/10.1214/ECP.v19-3641