### WEAK CONVERGENCE THEOREM FOR THE ERGODIC DISTRIBUTION OF THE RENEWAL-REWARD PROCESS WITH A GAMMA DISTRIBUTED INTERFERENCE OF CHANCE

**ROVSHAN ALIYEV, TAHIR KHANIYEV, AND NURGUL OKUR BEKAR**

*Theory of Stochastic Processes*

*Vol.15 (31), no.2, 2009, pp.42-53*

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.