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

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

Electronic Communications in Probability 13, pp. 99-112.

**ISSN/ISBN:** 1083-589X
**DOI:** 10.1214/ECP.v13-1358

**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:**

```
@article{,
author = "Dümbgen, Lutz and Leuenberger, Christoph",
doi = "10.1214/ECP.v13-1358",
fjournal = "Electronic Communications in Probability",
journal = "Electron. Commun. Probab.",
pages = "99--112",
pno = "10",
publisher = "The Institute of Mathematical Statistics and the Bernoulli Society",
title = "Explicit Bounds for the Approximation Error in Benford's Law",
url = "https://doi.org/10.1214/ECP.v13-1358",
volume = "13",
year = "2008"
}
```

**Reference Type:** Journal Article

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