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 Probability Surveys > Vol. 11 (2014) open journal systems 


Distribution of the sum-of-digits function of random integers: A survey

Louis H. Y. Chen, Department of Mathematics, National University of Singapore
Hsien-Kuei Hwang, Institute of Statistical Science, Institute of Information Science, Academia Si
Vytas Zacharovas, Dept. Mathematics & Informatics, Vilnius University


Abstract
We review some probabilistic properties of the sum-of-digits function of random integers. New asymptotic approximations to the total variation distance and its refinements are also derived. Four different approaches are used: a classical probability approach, Stein’s method, an analytic approach and a new approach based on Krawtchouk polynomials and the Parseval identity. We also extend the study to a simple, general numeration system for which similar approximation theorems are derived.

AMS 2000 subject classifications: Primary 60F05, 60C05; secondary 62E17, 11N37, 11K16.

Keywords: Sum-of-digits function, Stein’s method, Gray codes, total variation distance, numeration systems, Krawtchouk polynomials, digital sums, asymptotic normality

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Chen, Louis H. Y., Hwang, Hsien-Kuei, Zacharovas, Vytas, Distribution of the sum-of-digits function of random integers: A survey, Probability Surveys, 11, (2014), 177-236 (electronic). DOI: 10.1214/12-PS213.

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