Recurrence Relations and Benford's Law

TL;DR

This paper investigates how certain recurrence relations, including non-constant coefficient and higher-degree cases, produce sequences that follow Benford's law, extending previous fixed-coefficient results.

Contribution

It extends the understanding of Benford's law in recurrence sequences by proving new results for linear recurrences with non-constant coefficients and more complex forms.

Findings

Sequences from specific recurrence relations are Benford under certain conditions.

The behavior of these sequences can be analyzed via parameterization and equidistribution theory.

Generalizations include higher-degree and multiplicative recurrence relations, as well as random coefficient cases.

Abstract

There are now many theoretical explanations for why Benford's law of digit bias surfaces in so many diverse fields and data sets. After briefly reviewing some of these, we discuss in depth recurrence relations. As these are discrete analogues of differential equations and model a variety of real world phenomena, they provide an important source of systems to test for Benfordness. Previous work showed that fixed depth recurrences with constant coefficients are Benford modulo some technical assumptions which are usually met; we briefly review that theory and then prove some new results extending to the case of linear recurrence relations with non-constant coefficients. We prove that, for certain families of functions $f$ and $g$, a sequence generated by a recurrence relation of the form $a_{n+1} = f(n)a_n + g(n)a_{n-1}$ is Benford for all initial values. The proof proceeds by parameterizing the coefficients to obtain a recurrence relation of lower degree, and then converting to a new parameter space. From there we show that for suitable choices of $f$ and $g$ where $f(n)$ is nondecreasing and $g(n)/f(n)^2 \to 0$ as $n \to \infty$, the main term dominates and the behavior is equivalent to equidistribution problems previously studied. We also describe the results of generalizing further to higher-degree recurrence relations and multiplicative recurrence relations with non-constant coefficients, as well as the important case when $f$ and $g$ are values of random variables.