The above points can be justified by the following observations. First, as demonstrated in Fig. , the third order cumulant correction does improve the lineshape function in the short timescale limit, but introduces oscillations with amplitudes growing linearly in time, leading to an overestimation of oscillatory terms at longer timescales (see SI Secs. II, III and V). Therefore, a net effect of higher order cumulant terms must be a dampening of the linear divergence observed in the third order cumulant correction. This finding can be confirmed by evaluating higher order cumulants for model systems numerically (see SI Sec. III). Second, the timescale of divergences in the third order cumulant contribution, and thus the importance of higher order cumulant terms, is directly related to the degree of non-Gaussian behavior in the underlying energy gap fluctuations. Thus, we expect Φ to be a function of µ (3) and µ (4) .