Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T15:00:00.384Z Has data issue: false hasContentIssue false
  • English
  • Français

From uniform distributions to Benford's law

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Markov processes

Published online by Cambridge University Press:  14 July 2016

Élise Janvresse  and
Thierry de la Rue
Élise Janvresse*
Affiliation:
CNRS, Université de Rouen
Thierry de la Rue*
Affiliation:
CNRS, Université de Rouen
*
∗ Postal address: Université de Rouen, LMRS, UMR 6085 - CNRS, 76 821 Mont Saint Aignan, France
∗ Postal address: Université de Rouen, LMRS, UMR 6085 - CNRS, 76 821 Mont Saint Aignan, France
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

Benford's lawfirst-digit lawmantissauniform distributioncoupling method

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 11K99: None of the above, but in this section
Type
Short Communications
Information
Journal of Applied Probability , Volume 41 , Issue 4 , December 2004 , pp. 1203 - 1210
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Benford, F. (1938). The law of anomalous numbers. Proc. Amer. Phil. Soc. 78, 551–572.Google Scholar
Berger, A., Bunimovich, L., and Hill, T. (2005). One-dimensional dynamical systems and Benford's Law. Trans. Amer. Math. Soc. 357, 197–219.CrossRefGoogle Scholar
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.Google Scholar
Flehinger, B. J. (1966). On the probability that a random integer has initial digit. A. Amer. Math. Monthly 73, 1056–1061.Google Scholar
Hill, T. (1995). Base-invariance implies Benford's law. Proc. Amer. Math. Soc. 123, 887–895.Google Scholar
Hill, T. (1996). A statistical derivation of the significant-digit law. Statist. Sci. 10, 354–363.Google Scholar
Knuth, D. E. (1981). The Art of Computer Programming, Vol. 2. Addison-Wesley, Reading, MA.Google Scholar
Newcomb, S. (1881). Note on the frequency of use of the different digits in natural numbers. Amer. J. Math. 4, 39–40.Google Scholar
Pinkham, R. S. (1961). On the distribution of first significant digits. Ann. Math. Statist. 32, 1223–1230.Google Scholar
Raimi, R. A. (1976). The first digit problem. Amer. Math. Monthly 83, 521–538.CrossRefGoogle Scholar
Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Springer, New York.Google Scholar