Limit theorem for a semi-Markovian random walk with general interference of chance

Abstract

A semi-Markovian random-walk process with general interference of chance was constructed and investigated. The key point of this study is the assumption that the discrete interference of chance has a general form. Under some conditions, it is proved that the process is ergodic, and the exact forms of the ergodic distribution and characteristic function of the process are obtained. By using basic identity for random walks, the characteristic function of the process is expressed by the characteristic function of a boundary functional. Then, two-term asymptotic expansion for the characteristic function of the standardized process is found. Using this asymptotic expansion, a weak convergence theorem for the ergodic distribution of the standardized process is proved, and the limiting form for the ergodic distribution is obtained. The obtained limit distribution coincides with the limit distribution of the residual waiting time of the renewal process generated by a sequence of random variables expressing the discrete interference of chance.