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### Schatte, P (1988)

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

Probability Theory and Related Fields 77(2), 167-178.

ISSN/ISBN: 0178-8051 DOI: 10.1007/BF00334035

Abstract: 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.

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