Base-invariance implies Benford’s law

Hill, Theodore P.

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

Base-invariance implies Benford’s law
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by Theodore P. Hill PDF

Proc. Amer. Math. Soc. 123 (1995), 887-895 Request permission

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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