Notes on scale-invariance and base-invariance for Benford's Law

TL;DR

This paper characterizes Benford's law through generalized scale-invariance involving independent random variables and establishes conditions under which a positive continuous variable is Benford across multiple bases, using characteristic functions.

Contribution

It extends the concept of scale-invariance for Benford's law to random variables and provides a base-invariance characterization using characteristic functions.

Findings

Characterization of Benford's law via generalized scale-invariance.

Conditions under which a variable is Benford in multiple bases.

Use of characteristic functions to describe base-invariance.

Abstract

It is known that if X is uniformly distributed modulo 1 and Y is an arbitrary random variable independent of X then Y+X is also uniformly distributed modulo 1. We prove a converse for any continuous random variable Y (or a reasonable approximation to a continuous random variable) so that if X and Y+X are equally distributed modulo 1 and Y is independent of X then X is uniformly distributed modulo 1 (or approximates the uniform distribution equally reasonably). This translates into a characterization of Benford's law through a generalization of scale-invariance: from multiplication by a constant to multiplication by an independent random variable. We also show a base-invariance characterization: if a positive continuous random variable has the same significand distribution for two bases then it is Benford for both bases. The set of bases for which a random variable is Benford is characterized through characteristic functions.