Probability Theory and Related Fields

, Volume 77, Issue 2, pp 167–178 | Cite as

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

  • Peter Schatte


Let Zn 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](Zn) 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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© Springer-Verlag 1988

Authors and Affiliations

  • Peter Schatte
    • 1
  1. 1.Sektion Mathematik der Bergakademie FreibergFreibergGerman Democratic Republic

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