Open Access
2011 A basic theory of Benford’s Law
Arno Berger, Theodore P. Hill
Probab. Surveys 8: 1-126 (2011). DOI: 10.1214/11-PS175


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.


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Arno Berger. Theodore P. Hill. "A basic theory of Benford’s Law." Probab. Surveys 8 1 - 126, 2011.


Published: 2011
First available in Project Euclid: 28 July 2011

zbMATH: 1245.60016
MathSciNet: MR2846899
Digital Object Identifier: 10.1214/11-PS175

Primary: 11K06 , 37M10 , 39A60 , 60-01
Secondary: 37A45 , 60F15 , 62E10

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

Rights: Copyright © 2011 The Institute of Mathematical Statistics and the Bernoulli Society

Vol.8 • 2011
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