In climate model simulations of global warming, a number of statistics of convective rainfall are predicted to change. Here we present results for the Community Earth System Model (CESM), in which the threshold for the onset of tropical deep convection as a function of temperature and moisture is reasonably well simulated in current climate, even though it has never been previously validated for this property, and has a strong dependence on certain convective parameters. The way this threshold changes under global warming depends on the vertical profiles controlling convective conditional instability and exhibits a fairly simple dependence. The reasonable validation under current climate tends to boost confidence in the simulated changes, despite parameter sensitivity. Controlling for this threshold in computing probability density functions of precipitating points as a function of column water vapor and a bulk measure of tropospheric temperature is quite effective at collapsing these distributions for end-of-century climate under global warming onto those in the historical period. Differences are seen in increased probability of occurrence within the strong-convection regime and the largest precipitation event-size regime tends to show disproportionate sensitivity with warming.