# Base Dependence of Benford Random Variables

## Abstract

**:**

## 1. Introduction

## 2. Benford Random Variables

**Definition**

**1.**

**Proposition**

**1.**

**Proposition**

**2.**

**Example**

**1.**

**Example**

**2.**

## 3. The Benford Spectrum

**Definition**

**2.**

**Proposition**

**3**

**Proposition**

**4**

**Proof.**

**Proposition**

**5.**

**Proposition**

**6.**

**Proof.**

**Proposition**

**7.**

**Proposition**

**8**

## 4. Digression: Fourier Transforms

**Proposition**

**9**

## 5. A Framework for Benford Analysis

**Proposition**

**10.**

**Proof.**

**Proposition**

**11.**

**Proof.**

## 6. Base Dependence: Theory

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

**Example**

**6.**

## 7. Base Dependence: Examples

**Example**

**7.**

**Example**

**8.**

**Example**

**9.**

**Example**

**10.**

**Example**

**11.**

**Example**

**12.**

## 8. On “Base-Invariant Significant Digits”

**Proposition**

**12.**

**Proof.**

**Corollary**

**1.**

**Corollary**

**2.**

**Proposition**

**13.**

**Definition**

**3.**

**Proposition**

**14.**

**Proposition**

**15.**

**Proposition**

**16.**

**Proof.**

## 9. Conclusions and Prospect

- (1)
- Step functions that jump from 0 to 1 in a single step.
- (2)
- Increasing functions that are absolutely continuous.
- (3)
- Step functions that increase from 0 to 1 at a finite or countably infinite number of “points of jump.”
- (4)
- Convex combinations of seed functions in classes (2) and (3).
- (5)
- “Singular” distribution functions. These functions are increasing and continuous, but not absolutely continuous. The Cantor function is the best known example.
- (6)
- Seed functions satisfy a condition I call “unit interval increasing.” Every increasing function is unit interval increasing, but not conversely. That is, a function H may be unit interval increasing, but not everywhere increasing. Several examples of such seed functions are given in [2].

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. A Small Table of Fourier Transforms

No. | Name | Density $\mathit{g}\left(\mathit{x}\right)$ | Interval | Fourier Transform $\phantom{\rule{0.166667em}{0ex}}\widehat{\mathit{g}}\left(\mathit{\xi}\right)$ |
---|---|---|---|---|

1 | $N\left(\right)open="("\; close=")">0,1$ | ${\left(2\pi \right)}^{-1/2}{e}^{-{x}^{2}/2}$ | $\mathbb{R}$ | $exp\left(\right)open="("\; close=")">-2{\pi}^{2}{\xi}^{2}$ |

2 | $U\left(\right)open="["\; close="]">-a,a$ | $1/2a$ | $\left(\right)$ | $\frac{sin\left(\right)open="("\; close=")">2\pi a\xi}{}$ |

3 | $U\left(\right)open="["\; close="]">0,a$ | $1/a$ | $\left(\right)$ | $\frac{1-exp\left(\right)open="("\; close=")">-2\pi ia\xi}{}$ |

4 | Triangular | $\frac{1}{a}\left(\right)open="("\; close=")">1-\frac{\left|x\right|}{a}$ | $\left|x\right|\le a$ | $\frac{1-cos\left(\right)open="("\; close=")">2\pi a\xi}{}$ |

5 | Dual of 4 | $\frac{1-cos\left(\right)open="("\; close=")">2\pi ax}{}$ | $\mathbb{R}$ | $max\left(\right)open="("\; close=")">0,1-\frac{\left|\xi \right|}{a}$ |

6 | $\mathsf{\Gamma}\left(\right)open="("\; close=")">\alpha ,\beta $ | $\frac{1}{\mathsf{\Gamma}\left(\alpha \right){\beta}^{\alpha}}{x}^{\alpha -1}{e}^{-x/\beta}$ | $x>0$ | ${\left(\right)}^{1}$ |

7 | Laplace $\left(\right)$ | $\frac{1}{2}{e}^{-\left|x\right|}$ | $\mathbb{R}$ | $\frac{1}{1+4{\pi}^{2}{\xi}^{2}}$ |

8 | Cauchy $\left(\right)$ | $\frac{1}{\pi}\frac{1}{1+{x}^{2}}$ | $\mathbb{R}$ | ${e}^{-2\pi \left|\xi \right|}$ |

9 | Logistic $\left(\right)$ | ${\left(\right)}^{{e}^{x/2}}$ | $\mathbb{R}$ | $\frac{2{\pi}^{2}\xi}{sinh\left(\right)open="("\; close=")">2{\pi}^{2}\xi}$ |

## References

- Benford, F. The Law of Anomalous Numbers. Proc. Am. Philos. Soc.
**1938**, 78, 551–572. [Google Scholar] - Benford, F.A. Construction of Benford Random Variables: Generators and Seed Functions. arXiv
**2020**, arXiv:1609.04852. [Google Scholar] - Benford, F.A. Fourier Analysis and Benford Random Variables. arXiv
**2020**, arXiv:2006.07136. [Google Scholar] - Berger, A.; Theodore, H. An Introduction to Benford’s Law; Princeton University Press: Princeton, NJ, USA, 2015. [Google Scholar]
- Wójcik, M. Notes on Scale-Invariance and Base-Invariance for Benford’s Law. arXiv
**2013**, arXiv:1307.3620. [Google Scholar] - Whittaker, J. On Scale-Invariant Distributions. SIAM J. Appl. Math.
**1983**, 43, 257–267. [Google Scholar] [CrossRef] - Feller, W. An Introduction to Probability Theory and Its Applications, 2nd ed.; John Wiley & Sons: New York, NY, USA, 1971; Volume II. [Google Scholar]

Name | ${\mathit{h}}_{0}\left(\mathit{y}\right)$ | ${\widehat{\mathit{h}}}_{0}\left(\mathit{\xi}\right)$ |
---|---|---|

$N\left(\right)open="("\; close=")">0,\sigma $ | ${\left(2\pi {\sigma}^{2}\right)}^{-1/2}{e}^{-{y}^{2}/\left(\right)open="("\; close=")">2{\sigma}^{2}}$ | $exp\left(\right)open="("\; close=")">-2{\pi}^{2}{\sigma}^{2}{\xi}^{2}$ |

Laplace $\left(\right)$ | $\frac{1}{2\sigma}{e}^{-\left|y\right|/\sigma}$ | $\frac{1}{1+4{\pi}^{2}{\sigma}^{2}{\xi}^{2}}$ |

Cauchy $\left(\right)$ | $\frac{1}{\pi \sigma}{\left(\right)}^{1}-1$ | ${e}^{-2\pi \sigma \left|\xi \right|}$ |

Logistic $\left(\right)$ | $\frac{1}{\sigma}{\left(\right)}^{{e}^{y/\left(\right)open="("\; close=")">2\sigma}}+{e}^{-y/\left(\right)open="("\; close=")">2\sigma}-2$ | $\frac{2{\pi}^{2}\sigma \xi}{sinh\left(\right)open="("\; close=")">2{\pi}^{2}\sigma \xi}$ |

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Benford, F.
Base Dependence of Benford Random Variables. *Stats* **2021**, *4*, 578-594.
https://doi.org/10.3390/stats4030034

**AMA Style**

Benford F.
Base Dependence of Benford Random Variables. *Stats*. 2021; 4(3):578-594.
https://doi.org/10.3390/stats4030034

**Chicago/Turabian Style**

Benford, Frank.
2021. "Base Dependence of Benford Random Variables" *Stats* 4, no. 3: 578-594.
https://doi.org/10.3390/stats4030034