Base-invariance implies Benford’s law

Hill, Theodore P.

doi:10.1090/S0002-9939-1995-1233974-8

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: 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 $\sigma$ -algebra on the positive reals, and results for invariant measures on the circle.

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