  Serfozo (2009, Theorem 2.65) gives a useful central limit theorem for
processes with regenerative increments. Unfortunately, there is a gap in the
proof. We fill this gap, and at the same time we weaken the assumptions.
Furthermore, we give conditions for moment convergence in this setting. We give
also further results complementing results in Serfozo (2009) on the law of
large numbers and estimates for the mean; in particular, we show that there is
a gap between conditions for the weak and strong laws of large numbers.
