To ensure the safe delivery of proton therapy treatments it is important to evaluate the effect of potential uncertainties, such as patient mispositioning, on the intended dose distribution. However, it can be expected that the uncertainty resulting from patient positioning is reduced in a fractionated treatment due to the convergence of random variables with the delivery of repeated treatments. This is neglected by current approaches to robustness analysis resulting in an overly conservative assessment of the robustness which can lead to sub-optimal plans. Here, a fast method of accounting for this reduced uncertainty is presented. An estimated bound to the error in the dose distribution resulting from setup uncertainty over a specified number of fractions is calculated by considering the distribution of values for each voxel across 14 initial error scenarios. The bound on the error in a given voxel is estimated using a 99.9% confidence limit assuming a convergence towards a normal distribution in line with the central limit theorem, and a correction of [Formula: see text] accounting for the reduction in the standard deviation over n fractions. The proposed method was validated in 5 patients by comparison to Monte Carlo simulations of 300 treatment courses. A voxelwise and volumetric analysis of the estimated and simulated bounds to the uncertainty in the dose distribution demonstrate that the proposed technique can be used to assess proton plan robustness more accurately allowing for less conservative treatment plans.