### Becker, T, Burt, D, Corcoran, TC, Greaves-Tunnell, A, Iafrate, JR, Jing, J, Miller, SJ, Porfilio, JD, Ronan, R, Samranvedhya, J, Strauch, FW and Talbut, B (2018)

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

Annals of Physics 388, pp. 350–381.

**ISSN/ISBN:** Not available at this time.
**DOI:** 10.1016/j.aop.2017.11.013

**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. We handle this by using tools from Fourier analysis and irrationality exponents to obtain quantified convergence rates as well as introducing and developing techniques to measure and control the dependencies. The construction of these tools is one of the major motivations of this work, as our approach can be applied to many other dependent systems. As an example, we show that the n! entries in the determinant expansions of non
matrices with entries independently drawn from nice random variables converges to Benford’s Law.

**Bibtex:**

```
@ARTICLE{,
author = {Becker, Thealexa and Burt, David and Corcoran, Taylor C. and Greaves-Tunnell, Alec and
Iafrate, Joseph R. and Jing, Joy and {Miller}, Steven J. and Porfilio, Jaclyn D. and Ronan, Ryan and Samranvedhya, Jirapat and Strauch, Frederick W. and Talbut, Blaine},
title = {Benford's Law and Continuous Dependent Random Variables},
journal = {Annals of Physics},
year = 2018,
volume = {388},
pages = {350--381},
doi = {10.1016/j.aop.2017.11.013}.
}
```

**Reference Type:** Journal Article

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