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On a law of the iterated logarithm for sums mod 1 with application to Benford's law
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  • Published: February 1988

On a law of the iterated logarithm for sums mod 1 with application to Benford's law

  • Peter Schatte1 

Probability Theory and Related Fields volume 77, pages 167–178 (1988)Cite this article

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  • 14 Citations

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Summary

Let Z n be the sum mod 1 of n i.i.d.r.v. and let 1[0,x](·) be the indicator function of the interval [0, x]. Then the sequence 1[0,x](Z n ) does not converge for any x. But if arithmetic means are applied then under suitable suppositions convergence with probability one is obtained for all x as well-known. In the present paper the rate of this convergence is shown to be of order n -1/2 (loglogn)1/2 by using estimates of the remainder term in the CLT for m-dependent r.v.

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Authors and Affiliations

  1. Sektion Mathematik der Bergakademie Freiberg, Bernhard-von-Cotta-Str. 2, DDR-9200, Freiberg, German Democratic Republic

    Peter Schatte

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  1. Peter Schatte
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Schatte, P. On a law of the iterated logarithm for sums mod 1 with application to Benford's law. Probab. Th. Rel. Fields 77, 167–178 (1988). https://doi.org/10.1007/BF00334035

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  • Received: 03 June 1986

  • Revised: 20 October 1987

  • Issue Date: February 1988

  • DOI: https://doi.org/10.1007/BF00334035

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Keywords

  • Stochastic Process
  • Probability Theory
  • Statistical Theory
  • Indicator Function
  • Remainder Term
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