On the asymptotic behaviour of the semi-Markovian random walk with two barriers

SUMMARY On The Asimptotic Behaviour of The Semi-Markovian Random Walk With Two Barriers Many studies exist about random walk processes with two barriers in literature. These barriers have some properties like reflecting, delaying, absorbing etc. And these results which were found by those studies are theoretical and are not applicable for practice. Many asymptotic results that are very important for engineers and physicists can be found in our observing model. In this study, semi-Markovian random walk process X(t) with two barriers at 0 and P(>0), and the moment T, that process reaches the barriers, which is an important boundary functional for this process, are constructed mathematically and exact formulas are given for moments of random variable T,. Distribution function of X(t) process and its additive functional Jf (t) are explained by means of known probability characteristics of a {Tn} renewal process and {Sn} random walk process. Ergodic theorem is proved in very general conditions and ergodic distribution function of the process is obtained by probability characteristics of {Tn } and {Sn} process. Characteristic function of the ergodic distribution of the process is explained by N and SN boundary functional and exact expressions are found for its first and second moment. In case rj, random variable has two sided exponential distribution with A. > 0 parameter, the process is examined in detail. Weakly convergence theorem is also given for the process. Finally, two-termed asymptotic expansion is obtained for first-four moments of boundary functional N and SN, and ergodic distribution of the process in the case where rj, random variable has normal distribution by using simulation methods. Key Words: Stochastic process, Random walk process, Renewal process, Reward renewal process, Wald identity, Spitzer identity, Ladder variable, Wiener-Hopf. factorization, Weak convergence, Asymptotic expansion. VI