Probability Theory and Related Fields 77(2), 167-178.
ISSN/ISBN: 0178-8051 DOI: 10.1007/BF00334035
Abstract: 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.
Bibtex:
@article {MR927235,
AUTHOR = {Schatte, Peter},
TITLE = {On a law of the iterated logarithm for sums {${\rm mod}\,1$}
with application to {B}enford's law},
JOURNAL = {Probab. Theory Related Fields},
FJOURNAL = {Probability Theory and Related Fields},
VOLUME = {77},
YEAR = {1988},
NUMBER = {2},
PAGES = {167--178},
ISSN = {0178-8051},
CODEN = {PTRFEU},
MRCLASS = {60F15 (11K31)},
MRNUMBER = {927235 (89b:60081)},
MRREVIEWER = {L. Kuipers},
DOI = {10.1007/BF00334035},
URL = {http://dx.doi.org/10.1007/BF00334035},
}
Reference Type: Journal Article
Subject Area(s): Probability Theory