Weak convergence theorem for the ergodic distribution of a random walk with normal distributed interference of chance

Abstract

In this study, a semi-Markovian random walk process (X(t)) with a discrete interference of chance is investigated. Here, it is assumed that the ?n, n = 1, 2, 3, ..., which describe the discrete interference of chance are independent and identically distributed random variables having restricted normal distribution with parameters (a, ?2). Under this assumption, the ergodicity of the process X(t) is proved. Moreover, the exact forms of the ergodic distribution and characteristic function are obtained. Then, weak convergence theorem for the ergodic distribution of the process Wa(t) ? X(t)/a is proved under additional condition that ?/a ? 0 when a ? ?.