Probability Surveys

A basic theory of Benford’s Law

Arno Berger and Theodore P. Hill

Full-text: Open access

Abstract

Drawing from a large, diverse body of work, this survey presents a comprehensive and unified introduction to the mathematics underlying the prevalent logarithmic distribution of significant digits and significands, often referred to as Benford’s Law (BL) or, in a special case, as the First Digit Law. The invariance properties that characterize BL are developed in detail. Special attention is given to the emergence of BL in a wide variety of deterministic and random processes. Though mainly expository in nature, the article also provides strengthened versions of, and simplified proofs for, many key results in the literature. Numerous intriguing problems for future research arise naturally.

Article information

Source
Probab. Surveys Volume 8 (2011), 1-126.

Dates
First available in Project Euclid: 28 July 2011

Permanent link to this document
https://projecteuclid.org/euclid.ps/1311860830

Digital Object Identifier
doi:10.1214/11-PS175

Mathematical Reviews number (MathSciNet)
MR2846899

Zentralblatt MATH identifier
1245.60016

Subjects
Primary: 60-01: Instructional exposition (textbooks, tutorial papers, etc.) 11K06: General theory of distribution modulo 1 [See also 11J71] 37M10: Time series analysis 39A60: Applications
Secondary: 37A45: Relations with number theory and harmonic analysis [See also 11Kxx] 60F15: Strong theorems 62E10: Characterization and structure theory

Keywords
Benford’s Law significant digits uniform distribution mod 1 scale-invariance base-invariance sum-invariance shadowing difference equation random probability measure mixture of distributions

Citation

Berger, Arno; Hill, Theodore P. A basic theory of Benford’s Law. Probab. Surveys 8 (2011), 1--126. doi:10.1214/11-PS175. https://projecteuclid.org/euclid.ps/1311860830.


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