Benford Behavior in Stick Fragmentation Problems

Benford’s law is the statement that in many real-world data sets, the probability of having digit $d$ in base $B$, where $1\le d\le B$, as the first digit is ${\log }_B\left(\frac{d+1}{d}\right)$. 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.