On the Rate of Weak Convergence of Ergodic Distribution for a Renewal-Reward Process with a Generalized Reflecting Barrier Tahir Khaniyev (1) , Basak Gever(2) and Zulfiye Hanalioglu (3) (1) (3) TOBB University of Economics and Technology, Ankara, Turkey, [email protected] (2) TOBB University of Economics and Technology, Ankara, Turkey, [email protected] Karabuk University, Karabuk, Turkey, [email protected] Abstract In this study, a renewal-reward process (Xλ (t)) with a generalized reflecting barrier is constructed mathematically and under some weak conditions, the ergodicity of the process is proved. The explicit form of the ergodic distribution is found and after standardization, it is shown that the ergodic distribution converges to the limit distribution R(x), when λ → ∞, i.e., Z xZ ∞ 2 QX (λx) ≡ lim P {Xλ (t) ≤ λx} → R(x) ≡ [1 − F (u)]dudv, t→∞ m2 0 v where m2 ≡ E(η12 ) and F (x) is the distribution function of the initial random variables {ηn } (n = 1, 2, . . . ), which express the amount of rewards. Finally, to evaluate rate of the weak convergence, the following inequality is obtained: C |QX (λx) − R(x)| ≤ H(R(x)), λ where C is a positive constant and H(R(x)) is a certain function of the limit distribution R(x). Keywords: Renewal-reward process, reflecting barrier, ergodic distribution, weak convergence, rate of convergence. References [1] W. Feller, An Introduction to Probability Theory and Its Applications II, John Wiley, New York, 1971. [2] I.I. Gihman and A.V. Skorohod, Theory of Stocahstic Process II, Springer - Verlag, New York, 1975. [3] M. Brown and H. Solomon, A second - order approximation for the variance of a renewal - reward process, Stochastic Processes and Applications. 3 (1975), 301–314.