### Dümbgen, L and Leuenberger, C (2008)

#### Explicit Bounds for the Approximation Error in Benford’s Law

Electronic Communications in Probability 13, 99-112.

**ISSN/ISBN:** 1083-589X
**DOI:** Not available at this time.

**Abstract:** 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_{10} 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

**Bibtex:**

Not available at this time.

**Reference Type:** Journal Article

**Subject Area(s):** Probability Theory