We consider estimation of the cumulative mean function of a process recurring in time, such as the numbers of arrests or migrations accrued by an individual, as a function of their age. We call this the age pro le of a series of events. In some situations we can expect a nite value for the total number of events experienced by an individual, for example, when the distribution of the interevent times is improper, so that the process may cease at a nite time with positive probability. We propose and analyze a new estimator, constructed from Kaplan-Meier (KM) estimators of the interevent time distributions, for such an age pro le, and compare it with the Nelson-Aalen (NA) estimator of the cumulative mean function of a process. The KM estimator is proved to be uniformly consistent for the age pro le if and only if follow-up in the sample is suf cient in a sense that is manifested in practice by the leveling off of the pro les at their right-side ends, and is asymptotically normally distributed around its true value under mild conditions. Simulation results suggest that it is generally a better estimator than the NA estimator for the total number of events. It also appears to be more stable when applied to real data examples. For accurate estimation, it seems to be important to select a cohort of individuals whose ages at the rst event are as similar as possible. The estimators are illustrated on some time-to-arrest data.