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.
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