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Benford behavior resulting from stick and box fragmentation processes

Bruce Fang, Steven J. Miller

Abstract

Benford's law is the statement that in many real world data sets, the probability of having digit $d$ in base $B$ as the first digit is \log_{B}\!\left(\frac{d+1}{d}\right) for all $1 \leq d \leq B$. We sometimes refer to this as weak Benford behavior, and we say that a data set satisfies strong Benford behavior in base $B$ if the probability of having significand at most $s$ is \log_{B}\!\left(s\right) for all $1 \leq s < B$, . We examine Benford behaviors in two different probabilistic models: stick and box fragmentation models. Building on the work arXiv:1309.5603 on the single proportion stick fragmentation model, we employ combinatorial identities on multinomial coefficients to reduce the multi-proportion stick fragmentation model to the single proportion model. We then provide a necessary and sufficient condition for the lengths of the stick fragments to converge to strong Benford behavior along with a quantification of the discrepancy from uniform distribution on $[0,1]$ in terms of irrationality exponent. Then we answer a conjecture of arXiv:2304.08335 on the high-dimensional box fragmentation model. Using tools from Fourier analysis and 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.

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 in base as the first digit is \log_{B}\!\left(\frac{d+1}{d}\right) for all . We sometimes refer to this as weak Benford behavior, and we say that a data set satisfies strong Benford behavior in base if the probability of having significand at most is \log_{B}\!\left(s\right) for all , . We examine Benford behaviors in two different probabilistic models: stick and box fragmentation models. Building on the work arXiv:1309.5603 on the single proportion stick fragmentation model, we employ combinatorial identities on multinomial coefficients to reduce the multi-proportion stick fragmentation model to the single proportion model. We then provide a necessary and sufficient condition for the lengths of the stick fragments to converge to strong Benford behavior along with a quantification of the discrepancy from uniform distribution on in terms of irrationality exponent. Then we answer a conjecture of arXiv:2304.08335 on the high-dimensional box fragmentation model. Using tools from Fourier analysis and 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.

Paper Structure

This paper contains 22 sections, 17 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Previous Work on Fragmentation
  3. Proof of Theorem \ref{['multiproportion thm']}
  4. Preliminary
  5. Combinatorial identities
  6. Truncation
  7. Near uniform probability within small intervals
  8. Equidistribution within small intervals
  9. Evaluation of sums within and across intervals
  10. Proof of Theorem \ref{['box fragmentation conj answer arbitrary dim']}
  11. Preliminary
  12. PDF of $Y^{(N)}$
  13. Outline of problem
  14. Upper bound on error term $\mathcal{E}_N(a,b)$
  15. Strategy for remainder of proof
  16. ...and 7 more sections

Key Result

Proposition 1

Dia A sequence of random variables $\{X^{(N)}\}_{N=1}^\infty$ converges to strong Benford behavior in base $B$ if and only if for all $t\in [0,1)$. If uniform distribution characterization condition is satisfied, then we say that $\{\log_B(X^{(N)})\}_{N=1}^\infty$converges to being equidistributed mod 1.

Theorems & Definitions (33)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1: Uniform Distribution Characterization
  • Definition 4
  • Definition 5
  • Theorem 1
  • Theorem 2
  • Definition 6
  • Definition 7
  • ...and 23 more