Explicit bounds for the approximation error in Benford’s law

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) = log10(1 + 1/d) for d = 1, 2., 9. This phenomenon follows from another, maybe more intuitive fact, applied to Y:= log10 X: 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. © 2008 Applied Probability Trust.