In this work, we instead introduce and justify the novel hypothesis that while the exact nature of neglected higher order cumulants in the expansion cannot be known in arbitrary systems, some information of their net effect on the lineshape can be inferred. By quantifying the degree of non-Gaussian behavior in the energy gap fluctuations, through metrics such as skewness µ (3) and excess kurtosis µ (4) , one may approximately reintegrate the effects of the neglected cumulants in cancelling divergences of the lineshape. Specifically, we propose that an improved, and most importantly more stable, approximation to the exact lineshape compared to a pure third order cumulant expansion can be obtained through