Investigation of the semi-Markov processes with normal interference of chance by asymptotic methods

In this study, two important classes of the semi - Markov processes are considered.The first class, ?The Semi-Markovian Random Walk with a Normal Interference of Chance?, is constructed mathematically and the ergodicity of the process is proved under some weak conditions. Furthermore, exact expressions for the ergodic distribution function and the characteristic function are determined. Using these expressions the first four moments of the ergodic distribution are expressed by the first five moments of a certain boundary functional. Later, asymptotic expansions with three terms are found for the moments of the boundary functional. By using these results, the asymptotic expansions are proposed for various integrals related to moments of the boundary functional. Furthermore, asymptotic expansions with three terms for the first four moments of ergodic distribution of the process are found and the weak convergence theorem for the ergodic distribution of the process is proved.The second class, ?The Renewal - Reward Process with a Normal Interference of Chance?, is considered and the ergodicity of the process is proved in the second part of the study. Later, the ergodic distribution function of the process is expressed by means of a renewal function. Unlike the first class, the asymptotic expansions with three terms are determined for all moments of the ergodic distribution of the process. Moreover, weak convergence theorem for ergodic distribution of the process is proved and explicit form of the limit distribution is determined. Finally, the exact and asymptotic expressions are obtained for the moments of the boundary functionals of the process.