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 t
ime 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 conver
gence results based on the exact form of the tail of F, a typical concern i
n applications such as implications of long-range dependence in performance
of networking systems, reliability analysis or risk theory. (C) 1999 Elsev
ier Science B.V. All rights reserved.