We report our studies on exponentially-tempered Lévy sums that explain coherent random lasers based on nonresonant feedback. We investigate the hierarchy in the sums and identify the contribution of the extremes over a wide range of excitation energies and disorder strengths. Subsequently, we carry out experiments in which the physical manifestation of these extremes is revealed. At the appropriate gain and disorder, the extremes manifest as the sharp ultranarrow modes in the spectrum. At stronger excitation and disorder, the peaks disappear due to the reduced rarity of the extremes, compounded by the decreased magnitude effected by the tempering.