On the use of Lyapunov function methods in renewal theory

Based on recent results on the exploitation of “drift criteria” for general state-space Markov processes, we derive rates of convergence for (moments of ) processes associated with a renewal process with common inter-renewal time distribution F. Some of the results are classical and some are new, but the proofs are novel and, we believe, useful if one needs to derive convergence results based on the exact form of the tail of F, a typical concern in applications such as implications of long-range dependence in performance of networking systems, reliability analysis or risk theory.