Base-invariance implies Benford's law

Author:
Theodore P. Hill

Journal:
Proc. Amer. Math. Soc. **123** (1995), 887-895

MSC:
Primary 60A10; Secondary 28D05

DOI:
https://doi.org/10.1090/S0002-9939-1995-1233974-8

MathSciNet review:
1233974

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Abstract | References | Similar Articles | Additional Information

Abstract: A derivation of Benford's Law or the First-Digit Phenomenon is given assuming only base-invariance of the underlying law. The only base-invariant distributions are shown to be convex combinations of two extremal probabilities, one corresponding to point mass and the other a log-Lebesgue measure. The main tools in the proof are identification of an appropriate mantissa -algebra on the positive reals, and results for invariant measures on the circle.

**[1]**F. Benford,*The law of anomalous numbers*, Proc. Amer. Philos. Soc.**78**(1938), 551-572.**[2]**Daniel I. A. Cohen,*An explanation of the first digit phenomenon*, J. Combinatorial Theory Ser. A**20**(1976), no. 3, 367–370. MR**0406912****[3]**Richard Durrett,*Probability*, The Wadsworth & Brooks/Cole Statistics/Probability Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1991. Theory and examples. MR**1068527****[4]**William Feller,*An introduction to probability theory and its applications. Vol. I*, Third edition, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR**0228020****[5]**B. J. Flehinger,*On the probability that a random integer has initial digit 𝐴*, Amer. Math. Monthly**73**(1966), 1056–1061. MR**0204395**, https://doi.org/10.2307/2314636**[6]**Simon Newcomb,*Note on the Frequency of Use of the Different Digits in Natural Numbers*, Amer. J. Math.**4**(1881), no. 1-4, 39–40. MR**1505286**, https://doi.org/10.2307/2369148**[7]**Roger S. Pinkham,*On the distribution of first significant digits*, Ann. Math. Statist.**32**(1961), 1223–1230. MR**0131303**, https://doi.org/10.1214/aoms/1177704862**[8]**R. Raimi,*The peculiar distribution of first significant digits*, Sci. Amer.**221**(1969), 109-120.**[9]**Ralph A. Raimi,*The first digit problem*, Amer. Math. Monthly**83**(1976), no. 7, 521–538. MR**0410850**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1995-1233974-8

Keywords:
First-digit problem,
base-invariance,
scale-invariance,
Benford's Law,
invariant measure,
*n*th digit law

Article copyright:
© Copyright 1995
American Mathematical Society