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 Electronic Communications in Probability > Vol. 13 (2008) > Paper 10 open journal systems 


Explicit Bounds for the Approximation Error in Benford's Law

Lutz Dümbgen, University of Bern
Christoph Leuenberger, Ecole d'Ingénieurs de Fribourg


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 10X: For many real random variables Y , the remainder U := Y -⌊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 and powerful alternative to Fourier methods. As a by-product we obtain explicit bounds for the approximation error in Benford’s law.


Full text: PDF

Pages: 99-112

Published on: February 22, 2008
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Electronic Communications in Probability. ISSN: 1083-589X