Benford Behavior in Stick Fragmentation Problems

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.