Asymptotic expansions for the moments of a semi-Markovian random walk with exponential distributed interference of chance

In this paper, a semi-Markovian random walk process (X(t))(X(t)) with a discrete interference of chance is constructed and the ergodicity of this process is discussed. Some exact formulas for the first- and second-order moments of the ergodic distribution of the process X(t)X(t) are obtained, when the random variable ζ1ζ1 has an exponential distribution with the parameter λ>0λ>0. Here ζ1ζ1 expresses the quantity of a discrete interference of chance. Based on these results, the third-order asymptotic expansions for mathematical expectation and variance of the ergodic distribution of the process X(t)X(t) are derived, when λ→0λ→0.