Benford's Law and Continuous Dependent Random Variables

TL;DR

This paper investigates when data sets follow Benford's Law by analyzing models of particle decay and showing that dependent random variables in these models tend to Benford distribution as divisions increase.

Contribution

It provides a rigorous analysis of dependent random variables in decay models, proving convergence to Benford's Law with quantified rates.

Findings

Dependent variables in decay models tend to follow Benford's Law.

Convergence to Benford distribution is proven for many models.

Fourier analysis is used to quantify convergence rates.

Abstract

Many systems exhibit a digit bias. For example, the first digit base 10 of the Fibonacci numbers, or of $2^n$, equals 1 not 10% or 11% of the time, as one would expect if all digits were equally likely, but about 30% of the time. This phenomenon, known as Benford's Law, has many applications, ranging from detecting tax fraud for the IRS to analyzing round-off errors in computer science. The central question is determining which data sets follow Benford's law. Inspired by natural processes such as particle decay, our work examines models for the decomposition of conserved quantities. We prove that in many instances the distribution of lengths of the resulting pieces converges to Benford behavior as the number of divisions grow. The main difficulty is that the resulting random variables are dependent, which we handle by a careful analysis of the dependencies and tools from Fourier analysis to obtain quantified convergence rates.