Fourier Analysis and Benford Random Variables

TL;DR

This paper explores the properties of Benford random variables using Fourier analysis, providing both theoretical insights and an introduction to Fourier tools, highlighting their interconnections and applications in understanding digit distributions.

Contribution

It introduces a comprehensive Benford analysis framework for positive random variables and demonstrates the utility of Fourier series and transforms in this context.

Findings

Benford analysis reveals base dependence of digit distributions.

Fourier tools effectively characterize properties of Benford random variables.

The paper illustrates the connection between Fourier analysis and digit distribution phenomena.

Abstract

This paper has several major purposes. The central purpose is to describe the "Benford analysis" of a positive random variable and to summarize some results from investigations into base dependence of Benford random variables. The principal tools used to derive these results are Fourier series and Fourier transforms, and a second major purpose of this paper is to present an introductory exposition about these tools. My motivation for writing this paper is twofold. First, I think the theory of Benford random variables and the Benford analysis of a positive random variable are interesting and deserve to be better known. Second, I think that Benford analysis provides a really excellent illustration of the utility of Fourier series and transforms, and reveals certain interconnections between series and transforms that are not obvious from the usual way these subjects are introduced.