Explicit bounds for the approximation error in Benford's law

Details Explicit bounds for the approximation error in Benford's law Explicit bounds for the approximation error in Benford's law

2007-05-30

arxiv.org/abs/0705.4488v3

Electronic Communications in Probability 13 (2008), 99-112

Mathematics

Probability

16 pages, one figure

Scientific paper

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.

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