Regularity of digits and significant digits of random variables

Theodore P. Hill and Klaus Schürger

Stochastic Processes and their Applications, 2005, vol. 115, issue 10, 1723-1743

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.

Keywords: Normal; numbers; Significant; digits; Benford's; law; Digit-regular; random; variable; Significant-digit-regular; random; variable; Law; of; least; significant; digits; Floating-point; numbers; Nonleading; digits; Trailing; digits (search for similar items in EconPapers)
Date: 2005
References: View complete reference list from CitEc
Citations: View citations in EconPapers (4)

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