Limit theorems for stopping-times of random walks in a band

The aim of the paper is the study of control's impact on asymptotic behaviour and stability of the systems described by random walks with two absorbing boundaries. To this end the homogeneous (uncontrolled) random walk with three-valued jumps is considered at first. The object of investigation is the stopping-time $\eta_{x,n}$ where $x$ is the initial state and $n$ is the upper boundary, the lower one being 0. Next, it is shown that a two-level control policy radically changes the asymptotic behaviour of $\eta_{x,n}$, thus entailing the model's stability. For example, if the initial state $x$ tends to $\infty$, as $n\to\infty$, in such a way that $n-x\to\infty$, the limit distribution of the normalized random variable $\tau_{x,n}=\eta_{x,n}(\mathsf E\eta_{x,n})^{-1}$ is exponential with parameter 1 (independently of the jumps' mean value between two control levels $n_1$ and $n_2$). Whereas for the uncontrolled systems $\tau_{x,n}$ tends in probability to 1, as $n\to\infty$, if the jumps' mean is non-zero, the limit distribution of $\tau_{x,n}$ having a density $f_c(\cdot)$ for the case of zero mean, if $xn^{-1}\to c$, $0<c<1$, as $n\to\infty$. The main tool of investigation is the Laplace transform, which also gives the possibility to treat $\eta_{x,n}$ for initial states belonging to the “protection zones” in the neighbourhood of absorbing boundaries.