Generalized Continuous and Discrete Stick Fragmentation and Benford's Law

TL;DR

This paper extends basic stick fragmentation models to new variants and establishes precise conditions under which the resulting lengths follow Benford's law, including convergence criteria and counterexamples.

Contribution

It introduces three new fragmentation models, proves conditions for Benford's law adherence, and develops the concept of an aggregated limit for convergence.

Findings

Conditions for stick lengths to obey Benford's law are established.

The aggregated limit is introduced to ensure convergence in certain models.

Counterexamples show non-Benford behavior when conditions are not met.

Abstract

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.