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Precipitation accumulations, integrated over rainfall events, can be affected by both intensity and duration of the storm event. Thus, although precipitation intensity is widely projected to increase under global warming, a clear framework for predicting accumulation changes has been lacking, despite the importance of accumulations for societal impacts. Theory for changes in the probability density function (pdf) of precipitation accumulations is presented with an evaluation of these changes in global climate model simulations. We show that a simple set of conditions implies roughly exponential increases in the frequency of the very largest accumulations above a physical cutoff scale, increasing with event size. The pdf exhibits an approximately power-law range where probability density drops slowly with each order of magnitude size increase, up to a cutoff at large accumulations that limits the largest events experienced in current climate. The theory predicts that the cutoff scale, controlled by the interplay of moisture convergence variance and precipitation loss, tends to increase under global warming. Thus, precisely the large accumulations above the cutoff that are currently rare will exhibit increases in the warmer climate as this cutoff is extended. This indeed occurs in the full climate model, with a 3 degrees C end-of-century global-average warming yielding regional increases of hundreds of percent to >1,000% in the probability density of the largest accumulations that have historical precedents. The probabilities of unprecedented accumulations are also consistent with the extension of the cutoff.