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