Benford Behavior in Stick Fragmentation Problems

Bruce Fang (), Ava Irons, Ella Lippelman and Steven J. Miller
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Bruce Fang: Department of Mathematics, Williams College, Williamstown, MA 01267, USA
Ava Irons: Department of Mathematics, Williams College, Williamstown, MA 01267, USA
Ella Lippelman: Department of Mathematics and Computer Science, Colorado College, Colorado Spring, CO 80903, USA
Steven J. Miller: Department of Mathematics, Williams College, Williamstown, MA 01267, USA

Stats, 2025, vol. 8, issue 4, 1-10

Abstract: Benford’s law states that in many real-world datasets, the probability that the leading digit is d equals log 10 ( ( 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 log 10 ( 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.

Keywords: Benford’s law; high-dimensional fragmentation; multinomial distribution (search for similar items in EconPapers)
JEL-codes: C1 C10 C11 C14 C15 C16 (search for similar items in EconPapers)
Date: 2025
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