Geodesic
Weak convergence theorem for the ergodic distribution of the renewal-reward process with a gamma distributed interference of chance
Teoriâ slučajnyh processov, Tome 15 (2009) no. 2, pp. 42-53.

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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.
Keywords: Renewal-reward process, discrete interference of chance, asymptotic expansion, weak convergence, Monte-Carlo method.
Mots-clés : Laplace transform 
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Rovshan Aliyev; Tahir Khaniev; Nurgul Okur Bekar. Weak convergence theorem for the ergodic distribution of the renewal-reward process with a gamma distributed interference of chance. Teoriâ slučajnyh processov, Tome 15 (2009) no. 2, pp. 42-53. http://geodesic.mathdoc.fr/item/THSP_2009_15_2_a2/

[1] G. Alsmeyer, “Second-order approximations for certain stopped sums in extended renewal theory”, Advances in Applied Probability, 20 (1988), 391–410

[2] M. Brown, H. Solomon, “A second-order approximation for the variance of a renewal reward process”, Stochastic Processes and Applications, 3 (1975), 301–314

[3] A. Csenki, “Asymptotics for renewal-reward processes with retrospective reward structure”, Operation Research and Letters, 26 (2000), 201–209

[4] W. Feller, An Introduction to Probability Theory and Its Applications, v. II, Wiley, New York, 1971

[5] I. I. Gikhman, A. V. Skorohod, Theory of Stochastic Processes, v. II, Springer, Berlin, 1975

[6] J. B. Levy, M. S. Taqqu, “Renewal reward processes with heavy-tailed inter-renewal times and heavy-tailed rewards”, Bernoulli, 6:1 (2000), 23–44

[7] W. S. Jewell, “Fluctuation of a Renewal-Reward Process”, Journal of Mathematical Analysis and Applications, 19 (1967), 309–329

[8] T. A. Khaniyev, “About the moments of a generalized renewal process”, Transactions of NAS of Azerbaijan. Series of Physical Technical and Mathematical Sciences, 25:1 (2005), 95–100

[9] T. A. Khaniyev, T. Kesemen, R. T. Aliyev, A. Kokangul, “Asymptotic expansions for the moments of a semi-Markovian random walk with an exponential distributed interference of chance”, Statistics and Probability Letters, 78:6 (2008), 785–793

[10] S. M. Ross, Stochastic Processes, John Wiley Sons, New York, 1996