Probability Surveys

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

Louis H. Y. Chen, Hsien-Kuei Hwang, and Vytas Zacharovas

Full-text: Open access

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.

Article information

Source
Probab. Surveys, Volume 11 (2014), 177-236.

Dates
First available in Project Euclid: 10 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.ps/1412947841

Digital Object Identifier
doi:10.1214/12-PS213

Mathematical Reviews number (MathSciNet)
MR3269227

Zentralblatt MATH identifier
1327.60029

Subjects
Primary: 60F05: Central limit and other weak theorems 60C05: Combinatorial probability
Secondary: 62E17: Approximations to distributions (nonasymptotic) 11N37: Asymptotic results on arithmetic functions 11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. [See also 11A63]

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

Citation

Chen, Louis H. Y.; Hwang, Hsien-Kuei; Zacharovas, Vytas. Distribution of the sum-of-digits function of random integers: A survey. Probab. Surveys 11 (2014), 177--236. doi:10.1214/12-PS213. https://projecteuclid.org/euclid.ps/1412947841


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