On the semi-markovian random walk with special barriers

In this study, a semi-Markovian random walk with a discrete interference of chance (X(t)) is considered and under some weak assumptions the ergodicity of this process is discussed. The exact formulas for the first four moments of ergodic distribution of the process (X(t)) are obtained when the random variable ?_1, which is describing a discrete interference of chance, has a triangular distributionin the interval [s,S] with center (S+s)/2. Based on these results, the asymptotic expansions with three-term are obtained for the first four moments of the ergodic distribution of X(t), as a a?(S-s)/2??. Furthermore, the asymptotic expansions fort he variance, skewness and kurtosis of the ergodic distribution of the process X(t) are established. Finally, by using Monte Carlo experiments it is shown that the given approximating formulas provide high accuracy even for small values of parameter a. In second section of this study above mentioned all processes carried out for semi-Markovian random walk with a discrete interference of chance with delay.