Best Finite Approximations of Benford’s Law

  • Arno BergerEmail author
  • Chuang Xu


For arbitrary Borel probability measures with compact support on the real line, characterizations are established of the best finitely supported approximations, relative to three familiar probability metrics (Lévy, Kantorovich, and Kolmogorov), given any number of atoms, and allowing for additional constraints regarding weights or positions of atoms. As an application, best (constrained or unconstrained) approximations are identified for Benford’s Law (logarithmic distribution of significands) and other familiar distributions. The results complement and extend known facts in the literature; they also provide new rigorous benchmarks against which to evaluate empirical observations regarding Benford’s law.


Benford’s law Best uniform approximation Asymptotically best approximation Lévy distance Kantorovich distance Kolmogorov distance 

Mathematics Subject Classification (2010)

60B10 60E15 62E15 



The first author was partially supported by an Nserc Discovery Grant. Both authors gratefully acknowledge helpful suggestions made by F. Dai, B. Han, T.P. Hill, and an anonymous referee.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada

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