A correction term for the covariance of renewal-reward processes with multivariate rewards

We consider a renewal-reward process with multivariate rewards. Such a process is constructed from an i.i.d. sequence of time periods, to each of which there is associated a multivariate reward vector. The rewards in each time period may depend on each other and on the period length, but not on the other time periods. Rewards are accumulated to form a vector valued process that exhibits jumps in all coordinates simultaneously, only at renewal epochs.We derive an asymptotically exact expression for the covariance function (over time) of the rewards, which is used to refine a central limit theorem for the vector of rewards. As illustrated by a numerical example, this refinement can yield improved accuracy, especially for moderate time-horizons.