### Wojcik, MR (2013)

#### Notes on scale-invariance and base-invariance for Benford's Law

arXiv:1307.3620 [math.PR].

**ISSN/ISBN:** Not available at this time.
**DOI:** Not available at this time.

**Abstract:** It is known that if X is uniformly distributed modulo 1 and Y is an arbitrary random variable independent of X then Y+X is also uniformly distributed modulo 1. We prove a converse for any continuous random variable Y (or a reasonable approximation to a continuous random variable) so that if X and Y+X are equally distributed modulo 1 and Y is independent of X then X is uniformly distributed modulo 1 (or approximates the uniform distribution equally reasonably). This translates into a characterization of Benford's law through a generalization of scale-invariance: from multiplication by a constant to multiplication by an independent random variable.
We also show a base-invariance characterization: if a positive continuous random variable has the same significand distribution for two bases then it is Benford for both bases. The set of bases for which a random variable is Benford is characterized through characteristic functions.

**Bibtex:**

```
@unpublished{,
title={Notes on scale-invariance and base-invariance for Benford's Law},
author={Wojcik, Michal Ryszard},
eprint = {arXiv:1307.3620 [math.PR]},
year={2013}
note = {last accessed 6/20/2014},
url = {http://arxiv.org/abs/1307.3620},
}
```

**Reference Type:** E-Print

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