Benford Behavior in Stick Fragmentation Problems

TL;DR

This paper investigates how stick fragmentation processes exhibit Benford's law, providing conditions under which the fragment lengths follow strong Benford behavior using combinatorial identities and model reduction techniques.

Contribution

It introduces a method to reduce high-dimensional stick fragmentation models to a 1-dimensional case and establishes necessary and sufficient conditions for strong Benford behavior.

Findings

Fragment lengths can converge to strong Benford behavior.

A reduction technique simplifies high-dimensional models.

Necessary and sufficient conditions for Benford conformity are provided.

Abstract

Benford's law is the statement that in many real-world data sets, the probability of having digit \(d\) in base \(B\), where \(1 \leq d \leq B\), as the first digit is \(\log_{B}\left(\tfrac{d+1}{d}\right)\). We sometimes refer to this as weak Benford behavior, and we say that a data set exhibits strong Benford behavior in base \(B\) if the probability of having significand at most \(s\), where \(s \in [1,B)\), is \(\log_{B}(s)\). We examine Benford behaviors in the stick fragmentation model. Building on the work on the 1-dimensional stick fragmentation model, we employ combinatorial identities on multinomial coefficients to reduce the high-dimensional stick fragmentation model to the 1-dimensional model and provide a necessary and sufficient condition for the lengths of the stick fragments to converge to strong Benford behavior.