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 ζn, 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, σ2 . 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 σ/a → 0 when a → ∞.