Journal of Applied Probability

From uniform distributions to Benford's law

Élise Janvresse and Thierry de la Rue

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.

Article information

J. Appl. Probab. Volume 41, Number 4 (2004), 1203-1210.

First available in Project Euclid: 30 November 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 11K99: None of the above, but in this section

Benford's law first-digit law mantissa uniform distribution coupling method


Janvresse, Élise; de la Rue, Thierry. From uniform distributions to Benford's law. J. Appl. Probab. 41 (2004), no. 4, 1203--1210. doi:10.1239/jap/1101840566.

Export citation


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