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.