The time-dependence of heating in solar active regions can be studied by analyzing the slope of the emission measure distribution cool-ward of the peak. In a previous study we showed that low-frequency heating can account for 0% to 77% of active region core emission measures. We now turn our attention to heating by a finite succession of impulsive events for which the timescale between events on a single magnetic strand is shorter than the cooling timescale. We refer to this scenario as a nanoflare train and explore a parameter space of heating and coronal loop properties with a hydrodynamic model. Our conclusions are: (1) nanoflare trains are consistent with 86% to 100% of observed active region cores when uncertainties in the atomic data are properly accounted for; (2) steeper slopes are found for larger values of the ratio of the train duration $Delta_H$ to the post-train cooling and draining timescale $Delta_C$, where $Delta_H$ depends on the number of heating events, the event duration and the time interval between successive events ($tau_C$); (3) $tau_C$ may be diagnosed from the width of the hot component of the emission measure provided that the temperature bins are much smaller than 0.1 dex; (4) the slope of the emission measure alone is not sufficient to provide information about any timescale associated with heating - the length and density of the heated structure must be measured for $Delta_H$ to be uniquely extracted from the ratio $Delta_H/Delta_C$.
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