Explicit bounds for the approximation error in Benford's law

TL;DR

This paper derives explicit bounds on the approximation error in Benford's law by analyzing the uniformity of the fractional part of the logarithm of positive random variables, offering a practical alternative to Fourier-based methods.

Contribution

It provides new explicit bounds for the uniformity approximation of the fractional part of log10(X), improving understanding of Benford's law error bounds with a method based on total variation.

Findings

Explicit bounds in terms of total variation of the density of Y

Alternative to Fourier methods for error estimation

Bounds applicable to a wide class of distributions

Abstract

Benford's law states that for many random variables X > 0 its leading digit D = D(X) satisfies approximately the equation P(D = d) = log_{10}(1 + 1/d) for d = 1,2,...,9. This phenomenon follows from another, maybe more intuitive fact, applied to Y := log_{10}(X): For many real random variables Y, the remainder U := Y - floor(Y) is approximately uniformly distributed on [0,1). The present paper provides new explicit bounds for the latter approximation in terms of the total variation of the density of Y or some derivative of it. These bounds are an interesting alternative to traditional Fourier methods which yield mostly qualitative results. As a by-product we obtain explicit bounds for the approximation error in Benford's law.