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.