Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Base-invariance implies Benford's law

Author: Theodore P. Hill
Journal: Proc. Amer. Math. Soc. 123 (1995), 887-895
MSC: Primary 60A10; Secondary 28D05
MathSciNet review: 1233974
Full-text PDF

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

References [Enhancements On Off] (What's this?)

  • [1] F. Benford, The law of anomalous numbers, Proc. Amer. Philos. Soc. 78 (1938), 551-572.
  • [2] D. Cohen, An explanation of the first digit phenomenon, J. Combin. Theory Ser. A 20 (1976), 367-370. MR 0406912 (53:10698)
  • [3] R. Durrett, Probability: Theory and examples, Wadsworth, Belmont, 1991. MR 1068527 (91m:60002)
  • [4] W. Feller, An introduction to probability theory and its applications, Vol. 2, 3rd edition, Wiley, New York, 1968. MR 0228020 (37:3604)
  • [5] B. Flehinger, On the probability that a random number has initial digit A, Amer. Math. Monthly 73 (1966), 1056-1061. MR 0204395 (34:4237)
  • [6] S. Newcomb, Note on the frequency of use of the different digits in natural numbers, Amer. J. Math. 4 (1881), 39-40. MR 1505286
  • [7] R. Pinkham, On the distribution of the first significant digits, Ann. Math. Statist. 32 (1961), 1223-1230. MR 0131303 (24:A1155)
  • [8] R. Raimi, The peculiar distribution of first significant digits, Sci. Amer. 221 (1969), 109-120.
  • [9] -, The first digit problem, Amer. Math. Monthly 83 (1976), 521-538. MR 0410850 (53:14593)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 60A10, 28D05

Retrieve articles in all journals with MSC: 60A10, 28D05

Additional Information

Keywords: First-digit problem, base-invariance, scale-invariance, Benford's Law, invariant measure, nth digit law
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society