In this paper, the moments of a class of reward processes defined on a discretetime semi-Markov process and the asymptotic behaviors of the corresponding empirical estimators have been investigated. Some known results concerning the asymptotic distribution and properties of semi-Markov kernel have been obtained by a different approach. By using the empirical estimator of the semi-Markov kernel and the mentioned approach, the estimators for the moments of the reward process have been introduced and their asymptotic properties have been established. As a consequence of the strong consistency and asymptotic normality, the confidence intervals have also been obtained. A numerical example illustrates the results.