On a law of the iterated logarithm for sums mod 1 with application to Benford's law (Q1089673)

Let \(Z_ n\) be the sum mod 1 of n i.i.d. r.v. and let \(1_{[0,x)}(\cdot)\) 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 is well-known. In the present paper the rate of this convergence is shown to be of order \(n^{-1/2}(\log \log n)^{1/2}\) by using estimates of the remainder term in the CLT for m-dependent r.v.