Explicit bounds for the approximation error in Benford's law
Probability
2008-06-25 v3
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.
Keywords
Cite
@article{arxiv.0705.4488,
title = {Explicit bounds for the approximation error in Benford's law},
author = {Lutz Duembgen and Christoph Leuenberger},
journal= {arXiv preprint arXiv:0705.4488},
year = {2008}
}