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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A law of the iterated logarithm for arithmetic functions


Authors: István Berkes and Michel Weber
Journal: Proc. Amer. Math. Soc. 135 (2007), 1223-1232
MSC (2000): Primary 60F15, 11A25; Secondary 60G50
Published electronically: September 26, 2006
MathSciNet review: 2262929
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Abstract: Let $ X,X_1,X_2,\ldots $ be a sequence of centered iid random variables. Let $ f(n)$ be a strongly additive arithmetic function such that $ \sum_{p < n}\tfrac{f^2(p)}{p}\to\infty$ and put $ A_n= \sum_{p < n}\tfrac{f(p)}{p}$. If $ \mathbf{E} X^2 <\infty$ and $ f$ satisfies a Lindeberg-type condition, we prove the following law of the iterated logarithm:

$\displaystyle \limsup_{N\to \infty}{\sum_{n=1}^N f(n) X_n \over A_N \sqrt{2 N \log \log N}}\buildrel{a.s.}\over{=} \Vert X\Vert _2.$

We also prove the validity of the corresponding weighted strong law of large numbers in $ L^1$.


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Additional Information

István Berkes
Affiliation: Institut für Statistik, Technische Universität Graz, Steyrergasse 17/IV, A-8010 Graz, Austria
Email: berkes@tugraz.at

Michel Weber
Affiliation: Mathématique (IRMA), Université Louis-Pasteur et C.N.R.S., 7 rue René Descartes, 67084 Strasbourg Cedex, France
Email: weber@math.u-strasbg.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08557-1
PII: S 0002-9939(06)08557-1
Keywords: Iterated logarithm, strong laws of large numbers, weighted sums of iid random variables, strongly additive functions
Received by editor(s): May 25, 2005
Received by editor(s) in revised form: October 27, 2005
Published electronically: September 26, 2006
Additional Notes: The first author’s research was supported by the Hungarian National Foundation for Scientific Research, Grants T043037, T037886 and K61052
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2006 American Mathematical Society