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.