A combinatorial approach is used to derive asymptotic expressions for arbitrary moments of cumulative vector renewal reward processes, as the time horizon t goes to infinity. The rewards accumulate up to the last time interval not including time t. The analysis hinges on an expression of the moments in terms of the cumulants of the underlying time renewal sequence, and is founded on the recognition that certain sets of random variables are conditionally exchangeable. Subsequently, Smith's asymptotic theory of cumulants is applied. The formulae thus obtained generalize all currently known asymptotic results. They promise to lead to sharper estimates of the probabilistic behavior of cumulative processes over a finite time horizon than currently permitted by central limit theorem based analysis.