On the Limiting Behavior of the Characteristic Function of the Ergodic Distribution of the Semi-Markov Walk with Two Boundaries

The semi-Markov walk $(X(t))$ with two boundaries at the levels 0 and $\beta >0$ is considered. The characteristic function of the ergodic distribution of the process $X(t)$ is expressed in terms of the characteristics of the boundary functionals $N(z)$ and $S_{N(z)}$, where $N(z)$ is the first moment of exit of the random walk $\{S_{n}\}$, $n\ge 1$, from the interval $(-z,\beta-z)$, $z\in [0,\beta]$. The limiting behavior of the characteristic function of the ergodic distribution of the process $W_{\beta}(t)=2X(t)/\beta-1$ as $\beta \to \infty$ is studied for the case in which the components of the walk ($\eta_{i}$) have a two-sided exponential distribution.