Generalized Continuous and Discrete Stick Fragmentation and Benford's Law

Inspired by the basic stick fragmentation model proposed by Becker et al. in arXiv:1309.5603v4, we consider three new versions of such fragmentation models, namely, continuous with random number of parts, continuous with probabilistic stopping, and discrete with congruence stopping conditions. In all of these situations, we state and prove precise conditions for the ending stick lengths to obey Benford's law when taking the appropriate limits. We introduce the aggregated limit, necessary to guarantee convergence to Benford's law in the latter two models. We also show that resulting stick lengths are non-Benford when our conditions are not met. Moreover, we give a sufficient condition for a distribution to satisfy the Mellin transform condition introduced in arXiv:0805.4226v2, yielding a large family of examples.