  Martingales constitute a basic tool in stochastic analysis; this paper
considers their application to counting processes. We use this tool to revisit
a renewal theorem and its extensions for various counting processes. We first
consider a renewal process as a pilot example, deriving a new semimartingale
representation that differs from the standard decomposition via the stochastic
intensity function. We then revisit Blackwell's renewal theorem, its
refinements and extensions. Based on these observations, we extend the
semimartingale representation to a general counting process, and give
conditions under which asymptotic behaviour similar to Blackwell's renewal
theorem holds.
