A multidimensional renewal theorem is proven which generalizes the well-known result of Pyke on Markov renewal processes. The result is a "natural" setting of the multidimensional renewal theorem, and has applications to age-dependent branching processes.
"A Multidimensional Renewal Theorem." Ann. Probab. 4 (4) 656 - 661, August, 1976. https://doi.org/10.1214/aop/1176996034