On the asymptotic behaviour of the covariance function of the rewards of a multivariate renewal-reward process

In this study, a renewal-reward process with multivariate rewards is investigated and an asymptotic expansion for the covariance function of the rewards is derived. The remainder term of the result in Patch et al. (2015) is sharpened from the o(1) to o(t−k). Moreover, asymptotic expansions as t→∞ with the remainder term o(t−k) for the mathematical expectation and variance of the renewal-reward process also are obtained.