Asymptotic approach for random walk process with general interference of chance

In this study, a random walk process with general interference of chance is investigated. The main feature of the examined process is that general formulas are obtained, except for some special cases. The process is mathematically constructed, after that the general ergodic theorem is proved and the ergodic distribution of the process is obtained. The exact form of the characteristic function of this distribution is found with the help of the basic identity of random walk. Then, exact expressions and three-term asymptotic expansions are found for the moments of the boundary function of the process. Two term asymptotic expansion for the characteristic function of the standardized process is found and the weak convergence theorem is proved by using this expansion. It is been proved that the ergodic distribution of the standardized process according to the continuity theorem for characteristic functions converges to 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. Exact expressions are found for the first five moments of the ergodic distribution. Three term asymptotic expansions for the first five ergodic moments and four term asymptotic expansions for the first four ergodic moments of the process are obtained with the help of asymptotic expansions for the boundary functionals of the process. Approximate formulas are found for the characteristics of the process such as coefficient of variation, kurtosis and skewness with obtained asymptotic expansions of ergodic moments. In addition, the results obtained in the literature for special cases are calculated for the discrete interference of chances with different distributions (Weibull, symmetric triangular, exponential, Gamma) with the help of found general asymptotic expressions in this study. Keywords: Random walk process, Ergodic distribution, Weak convergence, Boundary functional, Ergodic moment, Asymptotic expansion.