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On the nature of Benford's Law

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  • Gottwald, Georg A.
  • Nicol, Matthew

Abstract

We study multiplicative and affine sequences of real numbers defined byN(j+1)=ζ(j)N(j)+η(j),where {ζ(j)} and {η(j)} are sequences of positive real numbers (in the multiplicative case η(j)=0 for all j). We investigate the conditions under which the leading digits k of {N(j)} have the following probability distribution, known as Benford's Law, P(k)=log10((k+1)/k). We present two main results. First, we show that contrary to the usual assumption in the literature, {ζ(j)} does not necessarily need to come from a chaotic or independent random process for Benford's Law to hold. The multiplicative driving force may be a deterministic quasiperiodic or even periodic forcing. Second, we give conditions under which the distribution of the first digits of an affine process displays Benford's Law. Our proofs use techniques from ergodic theory.

Suggested Citation

  • Gottwald, Georg A. & Nicol, Matthew, 2002. "On the nature of Benford's Law," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 303(3), pages 387-396.
    • Handle: RePEc:eee:phsmap:v:303:y:2002:i:3:p:387-396
    DOI: 10.1016/S0378-4371(01)00497-6
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    References listed on IDEAS

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    1. Pietronero, L. & Tosatti, E. & Tosatti, V. & Vespignani, A., 2001. "Explaining the uneven distribution of numbers in nature: the laws of Benford and Zipf," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 293(1), pages 297-304.
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    Cited by:

    1. César Carrera, 2015. "Tracking exchange rate management in Latin America," Review of Financial Economics, John Wiley & Sons, vol. 25(1), pages 35-41, April.

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