Asymptotic results for the absorption times of random walks with a barrier

A sequence $R_k^{(n)}:=R_{k-1}^{(n)}+\xi_k1_{\left\{R_{k-1}^{(n)}+\xi_k<n\right\}}$, $k\in\mathbf{N}$, $R_0^{(n)}:=0$, is called a random walk with a barrier $n \in \mathbf{N}$, where the $\xi_k$ are independent copies of a random variable $\xi$ assuming positive integer values. The asymptotic behavior of the absorption times is studied in the paper for a random walk with a barrier. This behavior depends on the properties of the tail of the distribution of the random variable $\xi$.