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From uniform distributions to Benford's law

  • Élise Janvresse (a1) and Thierry de la Rue (a1)
Abstract

We provide a new probabilistic explanation for the appearance of Benford's law in everyday-life numbers, by showing that it arises naturally when we consider mixtures of uniform distributions. Then we connect our result to a result of Flehinger, for which we provide a shorter proof and the speed of convergence.

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Corresponding author
Postal address: Université de Rouen, LMRS, UMR 6085 - CNRS, 76 821 Mont Saint Aignan, France
∗∗ Email address: elise.janvresse@univ-rouen.fr
∗∗∗ Email address: thierry.de-la-rue@univ-rouen.fr
References
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[1] Benford, F. (1938). The law of anomalous numbers. Proc. Amer. Phil. Soc. 78, 551572.
[2] Berger, A., Bunimovich, L., and Hill, T. (2005). One-dimensional dynamical systems and Benford's Law. Trans. Amer. Math. Soc. 357, 197219.
[3] Ferrari, P. A., and Galves, A. (2000). Coupling and Regeneration for Stochastic Processes. Sociedad Venezolana de Matematicas. Available at http://www.ime.usp.br/∼pablo.
[4] Flehinger, B. J. (1966). On the probability that a random integer has initial digit. A. Amer. Math. Monthly 73, 10561061.
[5] Hill, T. (1995). Base-invariance implies Benford's law. Proc. Amer. Math. Soc. 123, 887895.
[6] Hill, T. (1996). A statistical derivation of the significant-digit law. Statist. Sci. 10, 354363.
[7] Knuth, D. E. (1981). The Art of Computer Programming, Vol. 2. Addison-Wesley, Reading, MA.
[8] Newcomb, S. (1881). Note on the frequency of use of the different digits in natural numbers. Amer. J. Math. 4, 3940.
[9] Pinkham, R. S. (1961). On the distribution of first significant digits. Ann. Math. Statist. 32, 12231230.
[10] Raimi, R. A. (1976). The first digit problem. Amer. Math. Monthly 83, 521538.
[11] Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Springer, New York.
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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
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