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Regularity of digits and significant digits of random variables

Author

Listed:
  • Hill, Theodore P.
  • Schürger, Klaus

Abstract

A random variable X is digit-regular (respectively, significant-digit-regular) if the probability that every block of k given consecutive digits (significant digits) appears in the b-adic expansion of X approaches b-k as the block moves to the right, for all integers b>1 and k[greater-or-equal, slanted]1. Necessary and sufficient conditions are established, in terms of convergence of Fourier coefficients, and in terms of convergence in distribution modulo 1, for a random variable to be digit-regular (significant-digit-regular), and basic relationships between digit-regularity and various classical classes of probability measures and normal numbers are given. These results provide a theoretical basis for analyses of roundoff errors in numerical algorithms which use floating-point arithmetic, and for detection of fraud in numerical data via using goodness-of-fit of the least significant digits to uniform, complementing recent tests for leading significant digits based on Benford's law.

Suggested Citation

  • Hill, Theodore P. & Schürger, Klaus, 2005. "Regularity of digits and significant digits of random variables," Stochastic Processes and their Applications, Elsevier, vol. 115(10), pages 1723-1743, October.
    • Handle: RePEc:eee:spapps:v:115:y:2005:i:10:p:1723-1743
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    Citations

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    Cited by:

    1. Schürger, Klaus, 2008. "Extensions of Black-Scholes processes and Benford's law," Stochastic Processes and their Applications, Elsevier, vol. 118(7), pages 1219-1243, July.
    2. Mr. Jesus R Gonzalez-Garcia & Mr. Gonzalo C Pastor Campos, 2009. "Benford’s Law and Macroeconomic Data Quality," IMF Working Papers 2009/010, International Monetary Fund.
    3. Giuseppe Marzo & Yannick Tazzari & Stefano Bonnini, 2020. "Benford’s Law: genesi, letteratura e applicazioni empiriche," Working Papers 2020019, University of Ferrara, Department of Economics.
    4. Grendar, Marian & Judge, George & Schechter, Laura, 2007. "An empirical non-parametric likelihood family of data-based Benford-like distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 380(C), pages 429-438.

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