The performance of the truncated cumulant expansion can be rationalized when considering the lineshape function g(t) determining the spectrum (see Fig. )). The third order cumulant approximation yields a short timescale correction towards the exact lineshape function, causing an improvement in the placement and intensity of vibronic progressions in third-order spectra of mildly non-Gaussian systems (See SI Sec. II A). However, this short time correction causes a divergence in the real part of the lineshape function proportional to t as t → ∞, with additional divergent terms present in the imaginary part (See SI Sec. II and Ref. 39). By inspection of Eqn. 5, it may be noted that an exact lineshape function should not exhibit this divergence. Characteristics of the divergent terms, such as the fact that the real part of the lineshape function can become negative in the short timescale (see Fig. )) can be seen as the origin of spurious negative spectral features in the resulting absorption spectrum. Thus, this spurious asymptotic behavior is a clear target for seeking methodological improvements to the computation of cumulant-based spectra. If the goal is to produce the most accurate, physically applicable spec-trum possible, the inclusion of a correction that causes unphysical negative absorbances is undesirable, even if it leads to observed improvements in position and intensity of vibronic features. Therefore, it becomes of high interest to develop a methodology that retains the corrections afforded by the third order cumulant approach while reliably safeguarding against any unphysical spectral features. Ultimately, any technique developed must also be freely applicable to MD-type simulations of molecules in the condensed phase in order to be useful in realistic systems.