A law of the iterated logarithm for arithmetic functions
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- by István Berkes and Michel Weber PDF
- Proc. Amer. Math. Soc. 135 (2007), 1223-1232 Request permission
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: \[ \limsup _{N\to \infty }{\sum _{n=1}^N f(n) X_n \over A_N \sqrt {2 N \log \log N}}\buildrel {a.s.}\over {=} \|X\|_2.\] We also prove the validity of the corresponding weighted strong law of large numbers in $L^1$.References
- N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge, 1987. MR 898871, DOI 10.1017/CBO9780511721434
- P. D. T. A. Elliott, Probabilistic number theory. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 239, Springer-Verlag, New York-Berlin, 1979. Mean-value theorems. MR 551361
- P. Erdös and M. Kac, The Gaussian law of errors in the theory of additive number theoretic functions, Amer. J. Math. 62 (1940), 738–742. MR 2374, DOI 10.2307/2371483
- Evan Fisher, A Skorohod representation and an invariance principle for sums of weighted i.i.d. random variables, Rocky Mountain J. Math. 22 (1992), no. 1, 169–179. MR 1159950, DOI 10.1216/rmjm/1181072802
- H. Halberstam, On the distribution of additive number-theoretic functions, J. London Math. Soc. 30 (1955), 43–53. MR 66406, DOI 10.1112/jlms/s1-30.1.43
- Benton Jamison, Steven Orey, and William Pruitt, Convergence of weighted averages of independent random variables, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 4 (1965), 40–44. MR 182044, DOI 10.1007/BF00535481
- J. Kubilius, Probabilistic methods in the theory of numbers, Translations of Mathematical Monographs, Vol. 11, American Mathematical Society, Providence, R.I., 1964. MR 0160745
- Walter Philipp, Mixing sequences of random variables and probablistic number theory, Memoirs of the American Mathematical Society, No. 114, American Mathematical Society, Providence, R.I., 1971. MR 0437481
- Harold N. Shapiro, Distribution functions of additive arithmetic functions, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 426–430. MR 79609, DOI 10.1073/pnas.42.7.426
- Michel Weber, An ergodic theorem of arithmetic type, Tatra Mt. Math. Publ. 31 (2005), 123–129. MR 2208793
- Weber M., [2004] An LIL of arithmetical type, preprint.
- Michel Weber, An arithmetical property of Rademacher sums, Indag. Math. (N.S.) 15 (2004), no. 1, 133–149. MR 2061474, DOI 10.1016/S0019-3577(04)90011-0
- Michel Weber, On the order of magnitude of the divisor function, Acta Math. Sin. (Engl. Ser.) 22 (2006), no. 2, 377–382. MR 2214359, DOI 10.1007/s10114-005-0679-1
- Michel Weber, Divisors, spin sums and the functional equation of the zeta-Riemann function, Period. Math. Hungar. 51 (2005), no. 1, 119–131. MR 2180637, DOI 10.1007/s10998-005-0024-6
Additional Information
- István Berkes
- Affiliation: Institut für Statistik, Technische Universität Graz, Steyrergasse 17/IV, A-8010 Graz, Austria
- MR Author ID: 35400
- 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
- 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
- © Copyright 2006 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 135 (2007), 1223-1232
- MSC (2000): Primary 60F15, 11A25; Secondary 60G50
- DOI: https://doi.org/10.1090/S0002-9939-06-08557-1
- MathSciNet review: 2262929