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2012 Random number sequences and the first digit phenomenon
Bruno Massé, Dominique Schneider
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Electron. J. Probab. 17: 1-17 (2012). DOI: 10.1214/EJP.v17-1900


The sequences of mantissa of positive integers and of prime numbers are known not to be distributed as Benford's law in the sense of the natural density. We show that we can correct this defect by selecting the integers or the primes by means of an adequate random process and we investigate the rate of convergence. Our main tools are uniform bounds for deterministic and random trigonometric polynomials. We then adapt the random process to prove the same result for logarithms and iterated logarithms of integers. Finally we show that, in many cases, the mantissa law of the $n$th randomly selected term converges weakly to the Benford's law.


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Bruno Massé. Dominique Schneider. "Random number sequences and the first digit phenomenon." Electron. J. Probab. 17 1 - 17, 2012.


Accepted: 4 October 2012; Published: 2012
First available in Project Euclid: 4 June 2016

zbMATH: 1256.60005
MathSciNet: MR2988401
Digital Object Identifier: 10.1214/EJP.v17-1900

Primary: 60B10
Secondary: 11B05 , 11K99

Keywords: Benford's law , Density , mantissa , weak convergence


Vol.17 • 2012
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