The recently derived theoretical threshold function associated with the autoconversion process is generalized to account for the effect of the relative dispersion of the cloud droplet size distribution. This generalized threshold function theoretically demonstrates that the relative dispersion, which has been largely neglected to date, essentially controls the cloud-to-rain transition if the liquid water content and the droplet concentration are fixed. Comparison of the generalized threshold function to existing ad hoc threshold functions further reveals that the essential role of the spectral shape of the cloud droplet size distribution in rain initiation has been unknowingly buried in the arbitrary use of ad hoc threshold functions in atmospheric models such as global climate models, and that commonly used ad hoc threshold functions are unable to fully describe the threshold behavior of the autoconversion process that likely occurs in ambient clouds.