Benford behavior resulting from stick and box fragmentation processes

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(\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 $1 \leq s < B$, is \log_{B}!\left(s\right). We examine Benford behaviors in two different probabilistic models: stick and box fragmentation. Building on the work by arXiv:1309.5603 on the one-dimensional stick fragmentation model, we employ combinatorial identities on multinomial coefficients to reduce the high-dimensional stick fragmentation model to the one-dimensional model and provide a necessary and sufficient condition for the lengths of the stick fragments to converge to strong Benford behavior. Then we answer a conjecture by arXiv:2304.08335 on the high-dimensional box fragmentation model. Using tools from order statistics, we prove that under some mild conditions, faces of any arbitrary dimension of the box have total volume converging to strong Benford behavior.