@article{ECP1358,
author = {Lutz Dümbgen and Christoph Leuenberger},
title = {Explicit Bounds for the Approximation Error in Benford's Law},
journal = {Electron. Commun. Probab.},
fjournal = {Electronic Communications in Probability},
volume = {13},
year = {2008},
keywords = {Hermite polynomials, Gumbel distribution, Kuiper distance, normal distribution, total variation, uniform distribution, Weibull distribution},
abstract = {Benford's law states that for many random variables $X > 0$ its leading digit $D = D(X)$ satisfies approximately the equation $\mathbb{P}(D = d) = \log_{10}(1 + 1/d)$ for $d = 1,2,\ldots,9$. This phenomenon follows from another, maybe more intuitive fact, applied to $Y := \log_{10}X$: For many real random variables $Y$, the remainder $U := Y - \lfloor Y\rfloor$ 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 and powerful alternative to Fourier methods. As a by-product we obtain explicit bounds for the approximation error in Benford's law.},
pages = {no. 10, 99-112},
issn = {1083-589X},
doi = {10.1214/ECP.v13-1358},
url = {http://ecp.ejpecp.org/article/view/1358}}