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Regularity of digits and significant digits of random variables
Stochastic Processes and their Applications
  • Theodore P. Hill, Georgia Institute of Technology - Main Campus
  • Klaus Schürger, University of Bonn
Publication Date

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≥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.

Citation Information
Theodore P. Hill and Klaus Schürger. "Regularity of digits and significant digits of random variables" Stochastic Processes and their Applications Vol. 115 Iss. 10 (2005) p. 1723 - 1743
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