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Hill, TP and Schürger, K (2005)

Regularity of digits and significant digits of random variables

Journal of Stochastic Processes and their Applications 115(10), pp. 1723-1743.

ISSN/ISBN: 0304-4149 DOI: 10.1016/

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≥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.

@article {, AUTHOR = {Hill, Theodore P. and Sch{\"u}rger, Klaus}, TITLE = {Regularity of digits and significant digits of random variables}, JOURNAL = {Stochastic Process. Appl.}, FJOURNAL = {Stochastic Processes and their Applications}, VOLUME = {115}, YEAR = {2005}, NUMBER = {10}, PAGES = {1723--1743}, ISSN = {0304-4149}, DOI = {10.1016/}, URL = {}, }

Reference Type: Journal Article

Subject Area(s): Probability Theory