Benford Behavior in Stick Fragmentation Problems

Benford’s law states that in many real-world datasets, the probability that the leading digit is d equals log10((d+1)/d) for all 1≤d≤9. We call this weak Benford behavior. A dataset is said to follow strong Benford behavior if the probability that its significand (i.e., the significant digits in scientific notation) is at most s equals log10(s) for all s∈[1,10). We investigate Benford behavior in a multi-proportion stick fragmentation model, where a stick is split into m substicks according to fixed proportions at each stage. This generalizes previous work on the single proportion stick fragmentation model, where each stick is split into two substicks using one fixed proportion. We provide a necessary and sufficient condition under which the lengths of the stick fragments converge to strong Benford behavior in the multi-proportion model.