  Convergence results for averages of independent replications of counting
processes are established in a $p$-variation setting and under certain
assumptions. Such convergence results can be combined with functional
differentiability results in $p$-variation in order to study the asymptotic
properties of estimators that can be considered functionals of such averages of
counting processes. Examples of this are given in recurrent events settings,
confirming known results while also establishing the appropriateness of the
pseudo-observation method for regression analysis. In a recurrent events
setting with a terminal event, it is also established that it is more efficient
to discard complete information on a censoring time and instead consider the
censoring times censored by the terminal event.
