Weak convergence theorem for the ergodic distribution of the renewal-reward process with a gamma distributed interference of chance

In this study, a renewal-reward process with a discrete interference of chance $(X(t))$ is investigated. The ergodic distribution of this process is expressed by a renewal function. We assume that the random variables $\{\zeta _{n} \}$, $n\geq 1 $ which describe the discrete interference of chance form an ergodic Markov chain with the stationary gamma distribution with parameters $\left(\alpha ,\lambda \right)$, $\alpha>0 $, $\lambda>0 $. Under this assumption, an asymptotic expansion for the ergodic distribution of the stochastic process ${W}_{\lambda}\left({t}\right)=\lambda(X(t)-s)$ is obtained, as ${\lambda }\to 0$. Moreover, the weak convergence theorem for the process ${W}_{\lambda}\left({t}\right)$ is proved, and the exact expression of the limit distribution is derived. Finally, the accuracy of the approximation formula is tested by the Monte-Carlo simulation method.