Weak Convergence Theorem for Ergodic Distribution of Stochastic Processes with Discrete Interference of Chance and Generalized Reflecting Barrier

Abstract

In this paper, a stochastic process with discrete interference of chance and generalized reflecting barrier $\left(X\left(t\right)\right)$ is constructed and the ergodicity of this process is proved. Using basic identity for random walk processes, a characteristic function of the ergodic distribution is written with the help of characteristics of the boundary functional $S_{N_{1} (x)} $. Moreover, a weak convergence theorem for the ergodic distribution of the standardized process $Y_{\lambda} (t)\equiv X(t)/\lambda$ is proved, as $\lambda \to \infty $ and the limit form of the ergodic distribution is found.