### Becker, T, Corcoran, TC, Greaves-Tunnell, A, Iafrate, JR, Jing, J, Miller, SJ, Porfilio, JD, Ronan, R, Samranvedhya, J and Strauch, FW (2013)

#### Benford's Law and Continuous Dependent Random Variables

eprint arXiv:1309.5603, last revised 27 Dec 2013.

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

**Abstract:** Many mathematical, man-made and natural systems exhibit a leading-digit bias, where a first digit (base 10) of 1 occurs not 11% of the time, as one would expect if all digits were equally likely, but rather 30%. This phenomenon is known as Benford's Law. Analyzing which datasets adhere to Benford's Law and how quickly Benford behavior sets in are the two most important problems in the field. Most previous work studied systems of independent random variables, and relied on the independence in their analyses. Inspired by natural processes such as particle decay, we study the dependent random variables that emerge from models of decomposition of conserved quantities. We prove that in many instances the distribution of lengths of the resulting pieces converges to Benford behavior as the number of divisions grow, and give several conjectures for other fragmentation processes. The main difficulty is that the resulting random variables are dependent, which we handle by a careful analysis of the dependencies and tools from Fourier analysis to obtain quantified convergence rates. Our method can be applied to many other systems; as an example, we show that the $n!$ entries in the determinant expansions of $n\times n$ matrices with entries independently drawn from nice random variables converges to Benford's Law.

**Bibtex:**

```
@ARTICLE{,
author = {{Becker}, T. and {Corcoran}, T.~C. and {Greaves-Tunnell}, A. and
{Iafrate}, J.~R. and {Jing}, J. and {Miller}, S.~J. and {Porfilio}, J.~D. and
{Ronan}, R. and {Samranvedhya}, J. and {Strauch}, F.~W.},
title = "{Benford's Law and Continuous Dependent Random Variables}",
journal = {ArXiv e-prints},
archivePrefix = "arXiv",
eprint = {1309.5603},
primaryClass = "math.PR",
keywords = {Mathematics - Probability, 60A10, 11K06 (primary), (secondary) 60E10},
year = 2013,
month = sep,
adsurl = {http://adsabs.harvard.edu/abs/2013arXiv1309.5603B},
adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}
```

**Reference Type:** Preprint

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